awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
w :
  ∀ (x : (↑(↑Y : TopCat) : Type u_2)),
    Eq (α := (↑(↑Y : TopCat) : Type u_2))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
              TopCat.carrier) (X := (↑Y : TopCat)) (Y := (↑Y : TopCat))
              (Hom.base
                ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫
                  { base := H.hom, c := α.inv })) :
            (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
          x :
        (↑(↑Y : TopCat) : Type u_2))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
              TopCat.carrier) (Hom.base (𝟙 Y)) :
            (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
          x :
        (↑(↑Y : TopCat) : Type u_2))
h :
  ∀ (U : Opens (↑(↑Y : TopCat) : Type u_2)),
    Eq (α := Y.presheaf.obj (op U) ⟶ ((Opens.map (Hom.base (𝟙 Y))).op ⋙ Y.presheaf).obj (op U))
      ((CategoryStruct.comp (Y :=
            obj
              (Presheaf.pushforward C
                (Hom.base
                  ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫
                    { base := H.hom, c := α.inv : X.Hom Y })))
              Y.presheaf)
            (Hom.c
              ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv }))
            (whiskerRight (eqToHom ⋯) Y.presheaf)).app
        (op U))
      ((Hom.c (𝟙 Y)).app (op U))
⊢ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv } = 𝟙 Y

Tactic:
ext
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
x : (↑(↑Y : TopCat) : Type u_2)
w :
  (Hom.hom H.hom : (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
      ((Hom.hom H.inv : (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2)) x :
        (↑(↑X : TopCat) : Type u_2)) =
    x
⊢ Eq (α := (↑(↑Y : TopCat) : Type u_2))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC := TopCat.carrier)
            (X := (↑Y : TopCat)) (Y := (↑Y : TopCat))
            (Hom.base
              ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv })) :
          (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
        x :
      (↑(↑Y : TopCat) : Type u_2))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC := TopCat.carrier)
            (Hom.base (𝟙 Y)) :
          (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
        x :
      (↑(↑Y : TopCat) : Type u_2))

Tactic:
dsimp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
x : (↑(↑Y : TopCat) : Type u_2)
⊢ (Hom.hom H.hom : (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
      ((Hom.hom H.inv : (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2)) x :
        (↑(↑X : TopCat) : Type u_2)) =
    x

Tactic:
H.inv_hom_id_apply _

Premises used: CategoryTheory.Iso.inv_hom_id_apply
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
x : (↑(↑Y : TopCat) : Type u_2)
⊢ (Hom.hom H.hom : (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
      ((Hom.hom H.inv : (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2)) x :
        (↑(↑X : TopCat) : Type u_2)) =
    x

Tactic:
exact H.inv_hom_id_apply _

Premises used: CategoryTheory.Iso.inv_hom_id_apply
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
h :
  Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    ((α.inv.app (op U) ≫ (Presheaf.toPushforwardOfIso H α.hom).app (op ((Opens.map H.hom).obj U))) ≫
      Y.presheaf.map (NatTrans.app (eqToHom ⋯) (op U)))
    ((Hom.c (𝟙 Y)).app (op U))
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ ((Opens.map (Hom.base (𝟙 Y))).op ⋙ Y.presheaf).obj (op U))
    ((CategoryStruct.comp (Y :=
          obj
            (Presheaf.pushforward C
              (Hom.base
                ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫
                  { base := H.hom, c := α.inv : X.Hom Y })))
            Y.presheaf)
          (Hom.c
            ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv }))
          (whiskerRight (eqToHom ⋯) Y.presheaf)).app
      (op U))
    ((Hom.c (𝟙 Y)).app (op U))

Tactic:
dsimp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
h :
  Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    (α.inv.app (op U) ≫
      X.presheaf.map (eqToHom ⋯) ≫
        (obj (Presheaf.pushforward C H.hom) X.presheaf).map (NatTrans.app (eqToHom ⋯) (op U)) ≫
          α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    ((Hom.c (𝟙 Y)).app (op U))
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    ((α.inv.app (op U) ≫ (Presheaf.toPushforwardOfIso H α.hom).app (op ((Opens.map H.hom).obj U))) ≫
      Y.presheaf.map (NatTrans.app (eqToHom ⋯) (op U)))
    ((Hom.c (𝟙 Y)).app (op U))

Tactic:
simp only [Presheaf.toPushforwardOfIso_app, assoc, ← α.hom.naturality]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.NatTrans.naturality, TopCat.Presheaf.toPushforwardOfIso_app, CategoryTheory.Iso.hom
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
h :
  Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    (α.inv.app (op U) ≫ α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U))) ((Hom.c (𝟙 Y)).app (op U))
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    (α.inv.app (op U) ≫
      X.presheaf.map (eqToHom ⋯) ≫
        (obj (Presheaf.pushforward C H.hom) X.presheaf).map (NatTrans.app (eqToHom ⋯) (op U)) ≫
          α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    ((Hom.c (𝟙 Y)).app (op U))

Tactic:
simp only [eqToHom_map, eqToHom_app, eqToHom_trans_assoc, eqToHom_refl, id_comp]

Premises used: CategoryTheory.eqToHom_map, CategoryTheory.eqToHom_app, CategoryTheory.eqToHom_refl, CategoryTheory.eqToHom_trans_assoc, CategoryTheory.Category.id_comp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    (α.inv.app (op U) ≫ α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U))) ((Hom.c (𝟙 Y)).app (op U))

Tactic:
apply Iso.inv_hom_id_app

Premises used: CategoryTheory.Iso.inv_hom_id_app
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
w :
  ∀ (U : Opens (↑(↑Y : TopCat) : Type u_2)),
    Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
      (NatTrans.app (H.hom.c ≫ Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
      (NatTrans.app (𝟙 Y.presheaf) (op U))
⊢ H.hom.c ≫ Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c = 𝟙 Y.presheaf

Tactic:
ext U
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
w :
  Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
    (H.hom.c.app (op U) ≫ NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
    (NatTrans.app (𝟙 Y.presheaf) (op U))
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
    (NatTrans.app (H.hom.c ≫ Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
    (NatTrans.app (𝟙 Y.presheaf) (op U))

Tactic:
rw [NatTrans.comp_app]

Premises used: CategoryTheory.NatTrans.comp_app
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
    (H.hom.c.app (op U) ≫ NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
    (NatTrans.app (𝟙 Y.presheaf) (op U))

Tactic:
simpa using congr_arg (fun f => f ≫ eqToHom _) (congr_app H.inv_hom_id (op U))

Premises used: CategoryTheory.Iso.inv_hom_id, CategoryTheory.eqToHom, congr_arg, Opposite.op, CategoryTheory.CategoryStruct.comp, AlgebraicGeometry.PresheafedSpace.congr_app
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
w :
  ∀ (U : Opens (↑(↑Y : TopCat) : Type u_2)),
    Eq.{v_1 + 1} (α :=
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
      (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c ≫ H.hom.c) (op U) :
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
          (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
      (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (op U) :
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
          (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
⊢ Eq (α := obj (Presheaf.pushforward C H.hom.base) X.presheaf ⟶ obj (Presheaf.pushforward C H.hom.base) X.presheaf)
    (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c ≫ H.hom.c :
      obj (Presheaf.pushforward C H.hom.base) X.presheaf ⟶ obj (Presheaf.pushforward C H.hom.base) X.presheaf)
    (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf) :
      obj (Presheaf.pushforward C H.hom.base) X.presheaf ⟶ obj (Presheaf.pushforward C H.hom.base) X.presheaf)

Tactic:
ext U
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
w :
  Eq.{v_1 + 1} (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (op U) :
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
⊢ Eq.{v_1 + 1} (α :=
    (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c ≫ H.hom.c) (op U) :
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
    (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (op U) :
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))

Tactic:
dsimp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
w :
  Eq.{v_1 + 1} (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U)) :
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
⊢ Eq.{v_1 + 1} (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (op U) :
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))

Tactic:
rw [NatTrans.id_app]

Premises used: CategoryTheory.NatTrans.id_app
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
w :
  H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
⊢ Eq.{v_1 + 1} (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U)) :
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))

Tactic:
simp only [Presheaf.pushforwardToOfIso_app, Iso.symm_inv, mapIso_hom, forget_map,
      Iso.symm_hom, mapIso_inv, eqToHom_map, assoc]

Premises used: CategoryTheory.Category.assoc, TopCat.Presheaf.pushforwardToOfIso_app, CategoryTheory.eqToHom_map, CategoryTheory.Iso.symm_hom, CategoryTheory.Iso.symm_inv, AlgebraicGeometry.PresheafedSpace.forget_map, CategoryTheory.Functor.mapIso_inv, CategoryTheory.Functor.mapIso_hom
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))

Tactic:
have eq₁ := congr_app H.hom_inv_id (op ((Opens.map H.hom.base).obj U))
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
eq₁ :
  Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))
⊢ Eq (α :=
    Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (Y.presheaf.map (Quiver.Hom.op (eqToHom ⋯)) ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (Quiver.Hom.op (eqToHom ⋯)))

Tactic:
have eq₂ := H.hom.c.naturality (eqToHom (congr_obj (congr_arg Opens.map
      ((forget C).congr_map H.inv_hom_id.symm)) U)).op
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
eq₁ :
  Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))
eq₂ :
  Eq (α :=
    Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (Y.presheaf.map (Quiver.Hom.op (eqToHom ⋯)) ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (Quiver.Hom.op (eqToHom ⋯)))
⊢ Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    (H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
        H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj
          (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))))
    (eqToHom ⋯ :
      X.presheaf.obj ((Opens.map (Hom.base (𝟙 X))).op.obj (op ((Opens.map H.hom.base).obj U))) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))

Tactic:
rw [id_c, NatTrans.id_app, id_comp, eqToHom_map, comp_c_app] at eq₁

Premises used: CategoryTheory.eqToHom_map, AlgebraicGeometry.PresheafedSpace.comp_c_app, CategoryTheory.NatTrans.id_app, AlgebraicGeometry.PresheafedSpace.id_c, CategoryTheory.Category.id_comp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
eq₁ :
  Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((H.hom ≫ H.inv).c.app (op ((Opens.map H.hom.base).obj U)) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))
eq₂ :
  Eq (α :=
    Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (Y.presheaf.map (Quiver.Hom.op (eqToHom ⋯)) ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (Quiver.Hom.op (eqToHom ⋯)))
eq₃ :
  Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    (H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
        H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj
          (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))))
    (eqToHom ⋯ :
      X.presheaf.obj ((Opens.map (Hom.base (𝟙 X))).op.obj (op ((Opens.map H.hom.base).obj U))) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))
⊢ Eq (α :=
    Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (Y.presheaf.map (Quiver.Hom.op (eqToHom ⋯)) ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (Quiver.Hom.op (eqToHom ⋯)))

Tactic:
rw [id_c, NatTrans.id_app, id_comp, eqToHom_map, comp_c_app] at eq₁

Premises used: CategoryTheory.eqToHom_map, AlgebraicGeometry.PresheafedSpace.comp_c_app, CategoryTheory.NatTrans.id_app, AlgebraicGeometry.PresheafedSpace.id_c, CategoryTheory.Category.id_comp