Definitions/Def_PresheafOfModules_PullbackMonoidal.lean

import Mathlib

set_option autoImplicit false
set_option backward.isDefEq.respectTransparency false

universe v u v₁ u₁ v₂ u₂

open CategoryTheory MonoidalCategory Limits Functor.LaxMonoidal Functor.OplaxMonoidal

namespace PresheafOfModules

section restrictScalars

variable {C : Type u₁} [Category.{v₁} C] {R S : Cᵒᵖ ⥤ CommRingCat.{u}} (φ : S ⟶ R)

abbrev ringCatHom : S ⋙ forget₂ CommRingCat RingCat ⟶ R ⋙ forget₂ CommRingCat RingCat :=
  Functor.whiskerRight φ (forget₂ CommRingCat RingCat)

noncomputable abbrev restrictScalarsAt (U : Cᵒᵖ) : ModuleCat.{u} (R.obj U) ⥤ ModuleCat.{u} (S.obj U) :=
  ModuleCat.restrictScalars (φ.app U).hom

example (M : PresheafOfModules.{u} (R ⋙ forget₂ _ _)) (U : Cᵒᵖ) :
    ((restrictScalars (ringCatHom φ)).obj M).obj U = (restrictScalarsAt φ U).obj (M.obj U) := rfl

noncomputable def restrictScalarsε :
    𝟙_ (PresheafOfModules.{u} (S ⋙ forget₂ _ _)) ⟶
      (restrictScalars (ringCatHom φ)).obj (𝟙_ (PresheafOfModules.{u} (R ⋙ forget₂ _ _))) where
  app U := ε (restrictScalarsAt φ U)
  naturality {U V} i := by
    refine ModuleCat.hom_ext (LinearMap.ext fun (s : S.obj U) => ?_)
    change ε (restrictScalarsAt φ V) (S.map i s) = R.map i (ε (restrictScalarsAt φ U) s)
    erw [ModuleCat.restrictScalars_η, ModuleCat.restrictScalars_η]
    exact ConcreteCategory.congr_hom (φ.naturality i) s

noncomputable def restrictScalarsμ (M N : PresheafOfModules.{u} (R ⋙ forget₂ _ _)) :
    (restrictScalars (ringCatHom φ)).obj M ⊗ (restrictScalars (ringCatHom φ)).obj N ⟶
      (restrictScalars (ringCatHom φ)).obj (M ⊗ N) where
  app U := μ (restrictScalarsAt φ U) (M.obj U) (N.obj U)
  naturality {U V} i := by
    apply ModuleCat.MonoidalCategory.tensor_ext
    intro m n
    dsimp
    erw [Monoidal.tensorObj_map_tmul]
    erw [ModuleCat.restrictScalars_μ_tmul, ModuleCat.restrictScalars_μ_tmul]
    erw [Monoidal.tensorObj_map_tmul]
    rfl

noncomputable instance restrictScalars_laxMonoidal :
    (restrictScalars (ringCatHom φ)).LaxMonoidal where
  ε := restrictScalarsε φ
  μ M N := restrictScalarsμ φ M N
  μ_natural_left f N := by
    ext1 U
    simp only [comp_app, whiskerRight_app, restrictScalars_map_app]
    exact μ_natural_left (restrictScalarsAt φ U) (f.app U) (N.obj U)
  μ_natural_right M f := by
    ext1 U
    simp only [comp_app, whiskerLeft_app, restrictScalars_map_app]
    exact μ_natural_right (restrictScalarsAt φ U) (M.obj U) (f.app U)
  associativity M N P := by
    ext1 U
    simp only [comp_app, whiskerRight_app, whiskerLeft_app, restrictScalars_map_app,
      associator_hom_app]
    exact associativity (restrictScalarsAt φ U) (M.obj U) (N.obj U) (P.obj U)
  left_unitality M := by
    ext1 U
    simp only [comp_app, whiskerRight_app, restrictScalars_map_app, leftUnitor_hom_app]
    exact left_unitality (restrictScalarsAt φ U) (M.obj U)
  right_unitality M := by
    ext1 U
    simp only [comp_app, whiskerLeft_app, restrictScalars_map_app, rightUnitor_hom_app]
    exact right_unitality (restrictScalarsAt φ U) (M.obj U)

end restrictScalars

section pushforward

variable {C : Type u₁} [Category.{v₁} C] {D : Type u₂} [Category.{v₂} D] (F : C ⥤ D)
  {R : Dᵒᵖ ⥤ CommRingCat.{u}} {S : Cᵒᵖ ⥤ CommRingCat.{u}} (φ : S ⟶ F.op ⋙ R)

noncomputable instance pushforward_laxMonoidal :
    (pushforward (R := R ⋙ forget₂ _ _) (ringCatHom φ)).LaxMonoidal :=
  inferInstanceAs (pushforward₀OfCommRingCat F R ⋙
    restrictScalars (ringCatHom (R := F.op ⋙ R) φ)).LaxMonoidal

noncomputable instance pullback_oplaxMonoidal
    [(pushforward (R := R ⋙ forget₂ _ _) (ringCatHom φ)).IsRightAdjoint] :
    (pullback (R := R ⋙ forget₂ _ _) (ringCatHom φ)).OplaxMonoidal :=
  (pullbackPushforwardAdjunction (R := R ⋙ forget₂ _ _) (ringCatHom φ)).leftAdjointOplaxMonoidal

end pushforward

end PresheafOfModules

namespace CategoryTheory.Limits

lemma isIso_app_of_isColimit {𝒜 : Type*} [Category 𝒜] {ℬ : Type*} [Category ℬ]
    {J : Type*} [Category J] {H₁ H₂ : 𝒜 ⥤ ℬ} (τ : H₁ ⟶ H₂) {K : J ⥤ 𝒜} {c : Cocone K}
    (hc : IsColimit c) [PreservesColimit K H₁] [PreservesColimit K H₂]
    (h : ∀ j, IsIso (τ.app (K.obj j))) : IsIso (τ.app c.pt) := by
  let hc₁ := isColimitOfPreserves H₁ hc
  let hc₂ := isColimitOfPreserves H₂ hc
  let s : Cocone (K ⋙ H₂) :=
    { pt := H₁.obj c.pt
      ι := { app := fun j => inv (τ.app (K.obj j)) ≫ H₁.map (c.ι.app j)
             naturality := fun j j' g => by
               dsimp
               rw [← cancel_epi (τ.app (K.obj j))]
               simp only [Category.comp_id, IsIso.hom_inv_id_assoc]
               rw [← τ.naturality_assoc, IsIso.hom_inv_id_assoc, ← H₁.map_comp, Cocone.w] } }
  refine ⟨hc₂.desc s, ?_, ?_⟩
  · apply hc₁.hom_ext
    intro j
    have := hc₂.fac s j
    dsimp [s] at this ⊢
    rw [Category.comp_id, τ.naturality_assoc, this, IsIso.hom_inv_id_assoc]
  · apply hc₂.hom_ext
    intro j
    have := hc₂.fac s j
    dsimp [s] at this ⊢
    rw [Category.comp_id, reassoc_of% this, τ.naturality, IsIso.inv_hom_id_assoc]

end CategoryTheory.Limits

namespace PresheafOfModules

variable {C : Type u₁} [Category.{v₁} C] {R : Cᵒᵖ ⥤ CommRingCat.{u}}

local notation "R'" => R ⋙ forget₂ CommRingCat RingCat

lemma freeObj_map_freeMk (T : Cᵒᵖ ⥤ Type u) {X Y : Cᵒᵖ} (f : X ⟶ Y) (x : T.obj X) :
    (freeObj (R := R') T).map f (ModuleCat.freeMk x) = ModuleCat.freeMk (T.map f x) := by
  simp [freeObj, ModuleCat.freeDesc_apply]

noncomputable def freeεIso :
    𝟙_ (PresheafOfModules.{u} R') ≅ (free R').obj (𝟙_ (Cᵒᵖ ⥤ Type u)) :=
  isoMk (fun X => ModuleCat.FreeMonoidal.εIso (R.obj X)) (by
    intro X Y f
    refine ModuleCat.hom_ext (LinearMap.ext fun (r : R.obj X) => ?_)
    have h1 : ∀ (Z : Cᵒᵖ) (s : R.obj Z), (ModuleCat.FreeMonoidal.εIso (R.obj Z)).hom s =
        s • ModuleCat.freeMk PUnit.unit := fun Z s => by
      have := (ModuleCat.FreeMonoidal.εIso (R.obj Z)).hom.hom.map_smul s (1 : R.obj Z)
      rw [smul_eq_mul, mul_one] at this
      exact this
    change (ModuleCat.FreeMonoidal.εIso (R.obj Y)).hom ((unit R').map f r) =
      ((free R').obj (𝟙_ (Cᵒᵖ ⥤ Type u))).map f
        ((ModuleCat.FreeMonoidal.εIso (R.obj X)).hom r)
    rw [h1, h1, PresheafOfModules.map_smul]
    erw [freeObj_map_freeMk]
    rfl)

noncomputable def freeμIso (F G : Cᵒᵖ ⥤ Type u) :
    (free R').obj F ⊗ (free R').obj G ≅ (free R').obj (F ⊗ G) :=
  isoMk (fun X => ModuleCat.FreeMonoidal.μIso (R.obj X) (F.obj X) (G.obj X)) (by
    intro X Y f
    rw [← cancel_epi (ModuleCat.FreeMonoidal.μIso (R.obj X) (F.obj X) (G.obj X)).inv,
      Iso.inv_hom_id_assoc]
    apply ModuleCat.free_hom_ext
    rintro ⟨x, y⟩
    change (ModuleCat.FreeMonoidal.μIso (R.obj Y) (F.obj Y) (G.obj Y)).hom
      (((free R').obj F ⊗ (free R').obj G).map f
        ((ModuleCat.FreeMonoidal.μIso (R.obj X) (F.obj X) (G.obj X)).inv (ModuleCat.freeMk (x, y)))) =
      ((free R').obj (F ⊗ G)).map f (ModuleCat.freeMk (x, y))
    erw [ModuleCat.FreeMonoidal.μIso_inv_freeMk]
    erw [Monoidal.tensorObj_map_tmul]
    erw [freeObj_map_freeMk, freeObj_map_freeMk, freeObj_map_freeMk]
    erw [ModuleCat.FreeMonoidal.μIso_hom_freeMk_tmul_freeMk]
    rfl)

@[simp] lemma freeμIso_hom_app (F G : Cᵒᵖ ⥤ Type u) (X : Cᵒᵖ) :
    (freeμIso (R := R) F G).hom.app X =
      (ModuleCat.FreeMonoidal.μIso (R.obj X) (F.obj X) (G.obj X)).hom := rfl
@[simp] lemma freeεIso_hom_app (X : Cᵒᵖ) :
    (freeεIso (R := R)).hom.app X = (ModuleCat.FreeMonoidal.εIso (R.obj X)).hom := rfl

noncomputable instance free_monoidal : (free R').Monoidal :=
  Functor.CoreMonoidal.toMonoidal
    { εIso := freeεIso
      μIso := freeμIso
      μIso_hom_natural_left := fun {F F'} f G => by
        ext1 X
        simp only [comp_app, whiskerRight_app, free_map_app, freeμIso_hom_app,
          Monoidal.whiskerRight_app]
        exact μ_natural_left (ModuleCat.free (R.obj X)) (f.app X) (G.obj X)
      μIso_hom_natural_right := fun {F F'} G f => by
        ext1 X
        simp only [comp_app, whiskerLeft_app, free_map_app, freeμIso_hom_app,
          Monoidal.whiskerLeft_app]
        exact μ_natural_right (ModuleCat.free (R.obj X)) (G.obj X) (f.app X)
      associativity := fun F G H => by
        ext1 X
        simp only [comp_app, whiskerRight_app, whiskerLeft_app, free_map_app, freeμIso_hom_app,
          associator_hom_app, Monoidal.associator_hom_app]
        exact associativity (ModuleCat.free (R.obj X)) (F.obj X) (G.obj X) (H.obj X)
      left_unitality := fun F => by
        ext1 X
        simp only [comp_app, whiskerRight_app, free_map_app, freeμIso_hom_app, freeεIso_hom_app,
          leftUnitor_hom_app, Monoidal.leftUnitor_hom_app]
        exact left_unitality (ModuleCat.free (R.obj X)) (F.obj X)
      right_unitality := fun F => by
        ext1 X
        simp only [comp_app, whiskerLeft_app, free_map_app, freeμIso_hom_app, freeεIso_hom_app,
          rightUnitor_hom_app, Monoidal.rightUnitor_hom_app]
        exact right_unitality (ModuleCat.free (R.obj X)) (F.obj X) }

end PresheafOfModules

namespace PresheafOfModules

namespace PullbackMonoidal

open Opposite Functor.LaxMonoidal Functor.OplaxMonoidal

variable {C D : Type u} [SmallCategory C] [SmallCategory D]
  [HasFiniteProducts C] [HasFiniteProducts D]
  (F : C ⥤ D) [PreservesFiniteProducts F]
  {R : Dᵒᵖ ⥤ CommRingCat.{u}} {S : Cᵒᵖ ⥤ CommRingCat.{u}} (φ : S ⟶ F.op ⋙ R)

local notation "R'" => R ⋙ forget₂ CommRingCat RingCat
local notation "S'" => S ⋙ forget₂ CommRingCat RingCat

noncomputable abbrev pf : PresheafOfModules.{u} R' ⥤ PresheafOfModules.{u} S' :=
  pushforward (R := R') (ringCatHom φ)

noncomputable abbrev pb : PresheafOfModules.{u} S' ⥤ PresheafOfModules.{u} R' :=
  pullback (R := R') (ringCatHom φ)

noncomputable abbrev adj : pb F φ ⊣ pf F φ := pullbackPushforwardAdjunction (R := R') (ringCatHom φ)

noncomputable abbrev gS (U : C) : PresheafOfModules.{u} S' := (free S').obj (yoneda.obj U)
noncomputable abbrev gR (V : D) : PresheafOfModules.{u} R' := (free R').obj (yoneda.obj V)

noncomputable abbrev h₂ (U : C) :
    (pf F φ ⋙ coyoneda.obj (op (gS U))).CorepresentableBy (gR (F.obj U)) :=
  pushforwardCompCoyonedaFreeYonedaCorepresentableBy (R := R') (ringCatHom φ) U

noncomputable def u₂ (U : C) : gS U ⟶ (pf F φ).obj (gR (F.obj U)) :=
  (h₂ F φ U).homEquiv (𝟙 _)

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in
lemma freeYonedaEquiv_apply' {M : PresheafOfModules.{u} S'} {X : C}
    (f : gS X ⟶ M) : freeYonedaEquiv f = f.app (op X) (ModuleCat.freeMk (𝟙 X)) := rfl

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in
lemma gS_map_freeMk {U W W' : C} (a : W ⟶ U) (b : W' ⟶ W) :
    (gS (S := S) U).map b.op (ModuleCat.freeMk a) = ModuleCat.freeMk (b ≫ a) := by
  erw [freeObj_map_freeMk]; simp

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in
lemma gR_map_freeMk {U W W' : D} (a : W ⟶ U) (b : W' ⟶ W) :
    (gR (R := R) U).map b.op (ModuleCat.freeMk a) = ModuleCat.freeMk (b ≫ a) := by
  erw [freeObj_map_freeMk]; simp

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in

lemma free_map_app_freeMk {E : Type u} [SmallCategory E] {T : Eᵒᵖ ⥤ CommRingCat.{u}}
    {P Q : Eᵒᵖ ⥤ Type u} (ψ : P ⟶ Q) (X : Eᵒᵖ) (x : P.obj X) :
    ((free (T ⋙ forget₂ CommRingCat RingCat)).map ψ).app X (ModuleCat.freeMk x) =
      (ModuleCat.freeMk (ψ.app X x) : ((free (T ⋙ forget₂ CommRingCat RingCat)).obj Q).obj X) := by
  simp
  exact ModuleCat.free_map_apply _ _

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in

lemma freeμIso_inv_app_freeMk {E : Type u} [SmallCategory E] {T : Eᵒᵖ ⥤ CommRingCat.{u}}
    (P Q : Eᵒᵖ ⥤ Type u) (X : Eᵒᵖ) (z : (P ⊗ Q).obj X) :
    (freeμIso (R := T) P Q).inv.app X (ModuleCat.freeMk z) =
      (ModuleCat.freeMk z.1 ⊗ₜ[T.obj X] ModuleCat.freeMk z.2 :
        (((free (T ⋙ forget₂ CommRingCat RingCat)).obj P).obj X ⊗
          ((free (T ⋙ forget₂ CommRingCat RingCat)).obj Q).obj X :)) :=
  ModuleCat.FreeMonoidal.μIso_inv_freeMk _ z

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in

lemma tensorHom_app_tmul {E : Type u} [SmallCategory E] {T : Eᵒᵖ ⥤ CommRingCat.{u}}
    {M₁ M₂ M₃ M₄ : PresheafOfModules.{u} (T ⋙ forget₂ CommRingCat RingCat)}
    (f : M₁ ⟶ M₂) (g : M₃ ⟶ M₄) (X : Eᵒᵖ) (m : M₁.obj X) (n : M₃.obj X) :
    DFunLike.coe (α := (M₁.obj X ⊗ M₃.obj X :)) (β := fun _ => (M₂.obj X ⊗ M₄.obj X :))
      ((f ⊗ₘ g).app X).hom (m ⊗ₜ[T.obj X] n) = f.app X m ⊗ₜ[T.obj X] g.app X n := by
  erw [PresheafOfModules.tensorHom_app]
  exact ModuleCat.MonoidalCategory.tensorHom_tmul _ _ _ _

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in

lemma u₂_app_freeMk (U : C) {W : C} (a : W ⟶ U) :
    (u₂ F φ U).app (op W) (ModuleCat.freeMk a) =
      (ModuleCat.freeMk (F.map a) : (gR (R := R) (F.obj U)).obj (op (F.obj W))) := by
  have h1 : (u₂ F φ U).app (op U) (ModuleCat.freeMk (𝟙 U)) =
      (ModuleCat.freeMk (𝟙 (F.obj U)) : (gR (R := R) (F.obj U)).obj (op (F.obj U))) := by
    erw [freeYonedaEquiv_symm_app]; rfl
  have h2 := naturality_apply (u₂ F φ U) a.op (ModuleCat.freeMk (𝟙 U))
  rw [gS_map_freeMk, Category.comp_id] at h2
  rw [h2, h1]
  erw [pushforward_obj_map_apply]
  change (gR (R := R) (F.obj U)).map (F.map a).op (ModuleCat.freeMk (𝟙 (F.obj U))) = _
  rw [gR_map_freeMk, Category.comp_id]

noncomputable def e (U : C) : (pb F φ).obj (gS U) ≅ gR (F.obj U) where
  hom := ((adj F φ).homEquiv _ _).symm (u₂ F φ U)
  inv := (h₂ F φ U).homEquiv.symm ((adj F φ).unit.app (gS U))
  hom_inv_id := ((adj F φ).homEquiv _ _).injective (by
    rw [Adjunction.homEquiv_naturality_right, Equiv.apply_symm_apply]
    have := (h₂ F φ U).homEquiv_comp ((h₂ F φ U).homEquiv.symm ((adj F φ).unit.app (gS U))) (𝟙 _)
    rw [Category.id_comp, Equiv.apply_symm_apply] at this

    erw [Adjunction.homEquiv_unit, CategoryTheory.Functor.map_id, Category.comp_id]
    exact this.symm)
  inv_hom_id := (h₂ F φ U).homEquiv.injective (by
    rw [(h₂ F φ U).homEquiv_comp, Equiv.apply_symm_apply]
    change (adj F φ).unit.app (gS U) ≫ (pf F φ).map (((adj F φ).homEquiv _ _).symm (u₂ F φ U)) =
      (h₂ F φ U).homEquiv (𝟙 _)
    erw [← (adj F φ).homEquiv_unit, Equiv.apply_symm_apply]
    rfl)

lemma unit_comp_map_e_hom (U : C) :
    (adj F φ).unit.app (gS U) ≫ (pf F φ).map (e F φ U).hom = u₂ F φ U := by
  change (adj F φ).unit.app (gS U) ≫ (pf F φ).map (((adj F φ).homEquiv _ _).symm (u₂ F φ U)) = _
  erw [← (adj F φ).homEquiv_unit, Equiv.apply_symm_apply]

noncomputable def yonedaTensorIso {E : Type u} [SmallCategory E] [HasFiniteProducts E] (U V : E) :
    yoneda.obj U ⊗ yoneda.obj V ≅ yoneda.obj (U ⨯ V) :=
  NatIso.ofComponents (fun W => Equiv.toIso
    { toFun := fun p => prod.lift p.1 p.2
      invFun := fun c => (c ≫ prod.fst, c ≫ prod.snd)
      left_inv := fun p => Prod.ext (prod.lift_fst _ _) (prod.lift_snd _ _)
      right_inv := fun c => by apply Limits.prod.hom_ext <;> simp }) (by
    intro W W' f
    ext p
    change prod.lift ((yoneda.obj U ⊗ yoneda.obj V).map f p).1
        ((yoneda.obj U ⊗ yoneda.obj V).map f p).2 = f.unop ≫ prod.lift p.1 p.2
    erw [Monoidal.tensorObj_map, tensor_apply]
    apply Limits.prod.hom_ext <;> simp)

lemma yonedaTensorIso_inv_app_apply {E : Type u} [SmallCategory E] [HasFiniteProducts E] (U V : E)
    {W : E} (c : W ⟶ U ⨯ V) :
    (yonedaTensorIso U V).inv.app (op W) c = (c ≫ prod.fst, c ≫ prod.snd) := rfl

noncomputable def tensorUnitIso (E : Type u) [SmallCategory E] [HasFiniteProducts E] :
    𝟙_ (Eᵒᵖ ⥤ Type u) ≅ yoneda.obj (⊤_ E) :=
  NatIso.ofComponents (fun W => Equiv.toIso
    { toFun := fun _ => terminal.from _
      invFun := fun _ => PUnit.unit
      left_inv := fun _ => rfl
      right_inv := fun c => (terminal.hom_ext _ _) }) (by
    intro W W' f; ext p; apply terminal.hom_ext)

noncomputable def j (U V : C) : gS (S := S) U ⊗ gS V ≅ gS (U ⨯ V) :=
  freeμIso (yoneda.obj U) (yoneda.obj V) ≪≫ (free S').mapIso (yonedaTensorIso U V)

noncomputable def j₀ : 𝟙_ (PresheafOfModules.{u} S') ≅ gS (S := S) (⊤_ C) :=
  freeεIso ≪≫ (free S').mapIso (tensorUnitIso C)

noncomputable def r (U V : C) : gR (R := R) (F.obj (U ⨯ V)) ≅ gR (F.obj U) ⊗ gR (F.obj V) :=
  (free R').mapIso (yoneda.mapIso (PreservesLimitPair.iso F U V)) ≪≫
    ((free R').mapIso (yonedaTensorIso (F.obj U) (F.obj V))).symm ≪≫
    (freeμIso (yoneda.obj (F.obj U)) (yoneda.obj (F.obj V))).symm

noncomputable def r₀ : gR (R := R) (F.obj (⊤_ C)) ≅ 𝟙_ _ :=
  (free R').mapIso (yoneda.mapIso (PreservesTerminal.iso F)) ≪≫
    ((free R').mapIso (tensorUnitIso D)).symm ≪≫ freeεIso.symm

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in

lemma pf_μ_app_tmul (M N : PresheafOfModules.{u} R') (W : C)
    (m : M.obj (op (F.obj W))) (n : N.obj (op (F.obj W))) :
    DFunLike.coe (α := (((pf F φ).obj M).obj (op W) ⊗ ((pf F φ).obj N).obj (op W) :))
      (β := fun _ => ((pf F φ).obj (M ⊗ N)).obj (op W))
      ((μ (pf F φ) M N).app (op W)).hom (m ⊗ₜ[S.obj (op W)] n) = m ⊗ₜ[R.obj (op (F.obj W))] n := by
  change DFunLike.coe (α := (((pf F φ).obj M).obj (op W) ⊗ ((pf F φ).obj N).obj (op W) :))
      (β := fun _ => ((pf F φ).obj (M ⊗ N)).obj (op W))
    ((μ (restrictScalars (ringCatHom (R := F.op ⋙ R) φ))
        ((pushforward₀OfCommRingCat F R).obj M) ((pushforward₀OfCommRingCat F R).obj N) ≫
      (restrictScalars (ringCatHom (R := F.op ⋙ R) φ)).map
        (μ (pushforward₀OfCommRingCat F R) M N)).app (op W)).hom (m ⊗ₜ[S.obj (op W)] n) = _
  rw [show μ (pushforward₀OfCommRingCat F R) M N = 𝟙 _ from rfl, CategoryTheory.Functor.map_id,
    Category.comp_id]
  exact ModuleCat.restrictScalars_μ_tmul _ _ _ m n

lemma keyL (U V : C) :
    ((j (S := S) U V).inv ≫ ((u₂ F φ U ⊗ₘ u₂ F φ V) ≫
      μ (pf F φ) (gR (F.obj U)) (gR (F.obj V)))).app (op (U ⨯ V)) (ModuleCat.freeMk (𝟙 (U ⨯ V))) =
    (ModuleCat.freeMk (F.map prod.fst) ⊗ₜ[R.obj (op (F.obj (U ⨯ V)))]
      ModuleCat.freeMk (F.map prod.snd) :
      ((gR (R := R) (F.obj U)).obj (op (F.obj (U ⨯ V))) ⊗
        (gR (R := R) (F.obj V)).obj (op (F.obj (U ⨯ V))) :)) := by
  change (μ (pf F φ) (gR (F.obj U)) (gR (F.obj V))).app (op (U ⨯ V))
    (((u₂ F φ U ⊗ₘ u₂ F φ V).app (op (U ⨯ V)))
      ((freeμIso (yoneda.obj U) (yoneda.obj V)).inv.app (op (U ⨯ V))
        (((free S').map (yonedaTensorIso U V).inv).app (op (U ⨯ V))
          (ModuleCat.freeMk (𝟙 (U ⨯ V)))))) = _
  erw [free_map_app_freeMk, yonedaTensorIso_inv_app_apply, freeμIso_inv_app_freeMk]
  erw [tensorHom_app_tmul]
  erw [u₂_app_freeMk, u₂_app_freeMk]
  erw [pf_μ_app_tmul]
  simp only [Category.id_comp]

lemma keyR (U V : C) :
    (u₂ F φ (U ⨯ V) ≫ (pf F φ).map (r (R := R) F U V).hom).app (op (U ⨯ V))
      (ModuleCat.freeMk (𝟙 (U ⨯ V))) =
    (ModuleCat.freeMk (F.map prod.fst) ⊗ₜ[R.obj (op (F.obj (U ⨯ V)))]
      ModuleCat.freeMk (F.map prod.snd) :
      ((gR (R := R) (F.obj U)).obj (op (F.obj (U ⨯ V))) ⊗
        (gR (R := R) (F.obj V)).obj (op (F.obj (U ⨯ V))) :)) := by
  change ((pf F φ).map (r (R := R) F U V).hom).app (op (U ⨯ V))
    ((u₂ F φ (U ⨯ V)).app (op (U ⨯ V)) (ModuleCat.freeMk (𝟙 (U ⨯ V)))) = _
  erw [u₂_app_freeMk]
  erw [pushforward_map_app_apply]
  change (freeμIso (yoneda.obj (F.obj U)) (yoneda.obj (F.obj V))).inv.app (op (F.obj (U ⨯ V)))
    (((free R').map (yonedaTensorIso (F.obj U) (F.obj V)).inv).app (op (F.obj (U ⨯ V)))
      (((free R').map (yoneda.map (PreservesLimitPair.iso F U V).hom)).app (op (F.obj (U ⨯ V)))
        (ModuleCat.freeMk (F.map (𝟙 (U ⨯ V)))))) = _
  erw [free_map_app_freeMk, free_map_app_freeMk, yonedaTensorIso_inv_app_apply,
    freeμIso_inv_app_freeMk]
  change ModuleCat.freeMk ((F.map (𝟙 (U ⨯ V)) ≫ (PreservesLimitPair.iso F U V).hom) ≫ prod.fst)
      ⊗ₜ[R.obj (op (F.obj (U ⨯ V)))]
    ModuleCat.freeMk ((F.map (𝟙 (U ⨯ V)) ≫ (PreservesLimitPair.iso F U V).hom) ≫ prod.snd) = _
  simp only [CategoryTheory.Functor.map_id, Category.id_comp, Category.assoc,
    PreservesLimitPair.iso_hom, prodComparison_fst, prodComparison_snd]

lemma key_δ (U V : C) :
    (j (S := S) U V).inv ≫ ((u₂ F φ U ⊗ₘ u₂ F φ V) ≫
      μ (pf F φ) (gR (F.obj U)) (gR (F.obj V))) =
      u₂ F φ (U ⨯ V) ≫ (pf F φ).map (r (R := R) F U V).hom := by
  apply freeYonedaEquiv.injective
  rw [freeYonedaEquiv_apply', freeYonedaEquiv_apply', keyL, keyR]

lemma δ_gS_comp (U V : C) :
    δ (pb F φ) (gS U) (gS V) ≫ ((e F φ U).hom ⊗ₘ (e F φ V).hom) =
      (pb F φ).map (j (S := S) U V).hom ≫ (e F φ (U ⨯ V)).hom ≫ (r (R := R) F U V).hom := by
  apply ((adj F φ).homEquiv _ _).injective
  rw [Adjunction.homEquiv_naturality_right]
  erw [Adjunction.leftAdjointOplaxMonoidal_δ, Equiv.apply_symm_apply]
  rw [Category.assoc, ← Functor.LaxMonoidal.μ_natural, tensorHom_comp_tensorHom_assoc,
    unit_comp_map_e_hom, unit_comp_map_e_hom,
    Adjunction.homEquiv_naturality_left, Adjunction.homEquiv_naturality_right,
    Adjunction.homEquiv_unit, unit_comp_map_e_hom,
    ← cancel_epi (j (S := S) U V).inv, Iso.inv_hom_id_assoc]
  exact key_δ F φ U V

instance isIso_δ_gS (U V : C) : IsIso (δ (pb F φ) (gS U) (gS V)) := by
  have h : δ (pb F φ) (gS U) (gS V) =
      ((pb F φ).map (j (S := S) U V).hom ≫ (e F φ (U ⨯ V)).hom ≫ (r (R := R) F U V).hom) ≫
        inv ((e F φ U).hom ⊗ₘ (e F φ V).hom) := by
    rw [← δ_gS_comp, Category.assoc, IsIso.hom_inv_id, Category.comp_id]
  rw [h]
  infer_instance

omit [HasFiniteProducts C] [HasFiniteProducts D] [PreservesFiniteProducts F] in

lemma isIso_app_of_freeYoneda {ℬ : Type*} [Category ℬ]
    {H₁ H₂ : PresheafOfModules.{u} S' ⥤ ℬ}
    [PreservesColimitsOfSize.{u, u} H₁] [PreservesColimitsOfSize.{u, u} H₂] (τ : H₁ ⟶ H₂)
    (h : ∀ U : C, IsIso (τ.app (gS U))) (M : PresheafOfModules.{u} S') : IsIso (τ.app M) := by
  have hcop : ∀ (N : PresheafOfModules.{u} S'), IsIso (τ.app N.freeYonedaCoproduct) := fun N =>
    isIso_app_of_isColimit τ (coproductIsCoproduct (Elements.freeYoneda (M := N)))
      (fun m => h m.as.1.unop)
  have : PreservesColimitsOfSize.{0, 0} H₁ := preservesSmallestColimits_of_preservesColimits H₁
  have : PreservesColimitsOfSize.{0, 0} H₂ := preservesSmallestColimits_of_preservesColimits H₂
  exact isIso_app_of_isColimit τ M.isColimitFreeYonedaCoproductsCokernelCofork
    (by rintro (_ | _) <;> exact hcop _)

instance : PreservesColimitsOfSize.{u, u} (pb F φ) := (adj F φ).leftAdjoint_preservesColimits

instance isIso_δ (A B : PresheafOfModules.{u} S') : IsIso (δ (pb F φ) A B) := by

  have h1 : ∀ (U : C) (B : PresheafOfModules.{u} S'), IsIso (δ (pb F φ) (gS U) B) := fun U => by
    let σ : tensorLeft (gS (S := S) U) ⋙ pb F φ ⟶ pb F φ ⋙ tensorLeft ((pb F φ).obj (gS U)) :=
      { app := fun B => δ (pb F φ) (gS U) B
        naturality := fun B B' g => (δ_natural_right (pb F φ) (gS U) g).symm }
    exact isIso_app_of_freeYoneda (S := S) σ (fun V => isIso_δ_gS F φ U V)

  let τ : tensorRight B ⋙ pb F φ ⟶ pb F φ ⋙ tensorRight ((pb F φ).obj B) :=
    { app := fun A => δ (pb F φ) A B
      naturality := fun A A' f => (δ_natural_left (pb F φ) f B).symm }
  exact isIso_app_of_freeYoneda (S := S) τ (fun U => h1 U B) A

omit [HasFiniteProducts D] [PreservesFiniteProducts F] in
lemma keyL₀ : ((j₀ (S := S)).inv ≫ ε (pf F φ)).app (op (⊤_ C))
      (ModuleCat.freeMk (𝟙 (⊤_ C))) = (1 : R.obj (op (F.obj (⊤_ C)))) := by
  change (ε (pf F φ)).app (op (⊤_ C)) ((freeεIso (R := S)).inv.app (op (⊤_ C))
    (((free S').map (tensorUnitIso C).inv).app (op (⊤_ C)) (ModuleCat.freeMk (𝟙 (⊤_ C))))) = _
  erw [free_map_app_freeMk]
  erw [ModuleCat.FreeMonoidal.εIso_inv_freeMk]

  change (ε (restrictScalars (ringCatHom (R := F.op ⋙ R) φ)) ≫
      (restrictScalars (ringCatHom (R := F.op ⋙ R) φ)).map
        (ε (pushforward₀OfCommRingCat F R))).app (op (⊤_ C)) (1 : S.obj (op (⊤_ C))) = _
  rw [show ε (pushforward₀OfCommRingCat F R) = 𝟙 _ from rfl, CategoryTheory.Functor.map_id,
    Category.comp_id]
  erw [ModuleCat.restrictScalars_η]
  exact (φ.app (op (⊤_ C))).hom.map_one

lemma keyR₀ : (u₂ F φ (⊤_ C) ≫ (pf F φ).map (r₀ (R := R) F).hom).app (op (⊤_ C))
      (ModuleCat.freeMk (𝟙 (⊤_ C))) = (1 : R.obj (op (F.obj (⊤_ C)))) := by
  change ((pf F φ).map (r₀ (R := R) F).hom).app (op (⊤_ C))
    ((u₂ F φ (⊤_ C)).app (op (⊤_ C)) (ModuleCat.freeMk (𝟙 (⊤_ C)))) = _
  erw [u₂_app_freeMk]
  erw [pushforward_map_app_apply]
  change (freeεIso (R := R)).inv.app (op (F.obj (⊤_ C)))
    (((free R').map (tensorUnitIso D).inv).app (op (F.obj (⊤_ C)))
      (((free R').map (yoneda.map (PreservesTerminal.iso F).hom)).app (op (F.obj (⊤_ C)))
        (ModuleCat.freeMk (F.map (𝟙 (⊤_ C)))))) = _
  erw [free_map_app_freeMk, free_map_app_freeMk]
  exact ModuleCat.FreeMonoidal.εIso_inv_freeMk _ _

lemma key_η : (j₀ (S := S)).inv ≫ ε (pf F φ) = u₂ F φ (⊤_ C) ≫ (pf F φ).map (r₀ (R := R) F).hom := by
  apply freeYonedaEquiv.injective
  rw [freeYonedaEquiv_apply', freeYonedaEquiv_apply', keyL₀, keyR₀]

lemma η_eq : η (pb F φ) = (pb F φ).map (j₀ (S := S)).hom ≫ (e F φ (⊤_ C)).hom ≫ (r₀ (R := R) F).hom := by
  apply ((adj F φ).homEquiv _ _).injective
  erw [Adjunction.leftAdjointOplaxMonoidal_η, Equiv.apply_symm_apply]
  rw [Adjunction.homEquiv_naturality_left, Adjunction.homEquiv_naturality_right,
    Adjunction.homEquiv_unit, unit_comp_map_e_hom,
    ← cancel_epi (j₀ (S := S)).inv, Iso.inv_hom_id_assoc]
  exact key_η F φ

instance isIso_η : IsIso (η (pb F φ)) := by rw [η_eq]; infer_instance

noncomputable instance pullback_monoidal : (pb F φ).Monoidal := Functor.Monoidal.ofOplaxMonoidal _

end PullbackMonoidal

noncomputable instance pullback_monoidal' {C D : Type u} [SmallCategory C] [SmallCategory D]
    [HasFiniteProducts C] [HasFiniteProducts D] (F : C ⥤ D) [PreservesFiniteProducts F]
    {R : Dᵒᵖ ⥤ CommRingCat.{u}} {S : Cᵒᵖ ⥤ CommRingCat.{u}} (φ : S ⟶ F.op ⋙ R) :
    (pullback (R := R ⋙ forget₂ _ _) (ringCatHom φ)).Monoidal :=
  PullbackMonoidal.pullback_monoidal F φ

end PresheafOfModules