awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α := Set (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (Function.comp (β := (↑(↑(D.V (j, i)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(TopCat.Hom.hom (D.t i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (j, i)) : TopCat) : Type v)))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))
h₁ :
  Function.Bijective (α := (↑(↑(D.V (j, i)) : TopCat) : Type v)) (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (D.t j i).base) :
      (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
⊢ Eq (α := Set (↑(↑(D.V (j, i)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(TopCat.Hom.hom (D.t i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (j, i)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.preimage (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.t j i).base) :
        (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))

Tactic:
rw [Set.eq_preimage_iff_image_eq, ← Set.image_comp]

Premises used: Set.eq_preimage_iff_image_eq, Set.image_comp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Function.Bijective (α := (↑(↑(D.V (j, i)) : TopCat) : Type v)) (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (D.t j i).base) :
      (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))

Tactic:
CategoryTheory.ConcreteCategory.bijective_of_isIso (C := TopCat) _

Premises used: CategoryTheory.ConcreteCategory.bijective_of_isIso
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Function.Bijective (α := (↑(↑(D.V (j, i)) : TopCat) : Type v)) (β := (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (D.t j i).base) :
      (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))

Tactic:
exact CategoryTheory.ConcreteCategory.bijective_of_isIso (C := TopCat) _

Premises used: CategoryTheory.ConcreteCategory.bijective_of_isIso
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α := Set (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.t i j ≫ D.t j i).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))
⊢ Eq (α := Set (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (Function.comp (β := (↑(↑(D.V (j, i)) : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.t j i).base) :
          (a : (↑(↑(D.V (j, i)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
        (⇑(TopCat.Hom.hom (D.t i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (j, i)) : TopCat) : Type v)))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))

Tactic:
change (D.t i j ≫ D.t j i).base '' _ = _

Premises used: CategoryTheory.GlueData.t, CategoryTheory.CategoryStruct.comp, AlgebraicGeometry.PresheafedSpace.Hom.base, AlgebraicGeometry.PresheafedSpace.GlueData.toGlueData
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α := Set (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Hom.base (𝟙 (D.V (i, j))))) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))
⊢ Eq (α := Set (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.t i j ≫ D.t j i).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))

Tactic:
rw [𝖣.t_inv]
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Eq (α := Set (↑(↑(D.V (i, j)) : TopCat) : Type v))
    (Set.image (α := (↑(↑(D.V (i, j)) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Hom.base (𝟙 (D.V (i, j))))) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.V (i, j)) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (D.f i j).base) :
          (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (D.f i j).base) :
        (a : (↑(↑(D.V (i, j)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
      (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))

Tactic:
simp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Mono (X := (D.mapGlueData (forget C)).glued) (Y := (↑D.glued : TopCat))
    (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued)
⊢ Function.Injective (α := (↑(D.mapGlueData (forget C)).glued : Type v)) (β := (↑(↑D.glued : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (D.mapGlueData (forget C)).glued) (Y := (↑D.glued : TopCat))
            (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued) :
          C((↑(D.mapGlueData (forget C)).glued : Type v), (↑(↑D.glued : TopCat) : Type v))) :
      (a : (↑(D.mapGlueData (forget C)).glued : Type v)) → (↑(↑D.glued : TopCat) : Type v))

Tactic:
rw [← TopCat.mono_iff_injective]

Premises used: TopCat.mono_iff_injective
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Mono (X := (D.mapGlueData (forget C)).glued) (Y := (↑D.glued : TopCat))
    (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued)

Tactic:
infer_instance
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j k : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
erw [← comp_c_app, ← comp_c_app]

Premises used: AlgebraicGeometry.PresheafedSpace.comp_c_app
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j k : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
h :
  Eq (α :=
    (D.U i).presheaf.obj (op U) ⟶
      (Functor.obj (TopCat.Presheaf.pushforward C (D.f j k).base) (D.V (j, k)).presheaf).obj
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι j)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)))))
    (CategoryStruct.comp (Y :=
      (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k))))
            (pullback (D.f j i) (D.f j k)).presheaf).obj
        (op ((Opens.map (D.t j i).base).1 (unop ((Opens.map (D.f i j).base).op.obj (op U))))))
      (NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U) :
        (D.U i).presheaf.obj (op U) ⟶
          (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)))
                (pullback (D.f j i) (D.f j k)).presheaf).obj
            (op U))
      (invApp (pullback.snd (D.f j i) (D.f j k))
          ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k))) :
                Opens (↑(↑(D.V (j, i)) : TopCat) : Type v) ⥤
                  Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
            ((Opens.map (D.t j i).base).1 (unop ((Opens.map (D.f i j).base).op.obj (op U))))) ≫
        (D.V (j, k)).presheaf.map (eqToHom ⋯ ≫ (Opens.map (D.f j k).base).op.map (Quiver.Hom.op (eqToHom ⋯)))))
    (CategoryStruct.comp (Y :=
      (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)))
            (pullback (D.f j i) (D.f j k)).presheaf).obj
        (op U))
      (NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U))
      (invApp (pullback.snd (D.f j i) (D.f j k))
          ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
            U) ≫
        (D.V (j, k)).presheaf.map (eqToHom ⋯)))
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simp_rw [Category.assoc]

Premises used: CategoryTheory.Category.assoc
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j k : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Eq (α :=
    (D.U i).presheaf.obj (op U) ⟶
      (Functor.obj (TopCat.Presheaf.pushforward C (D.f j k).base) (D.V (j, k)).presheaf).obj
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι j)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v)))))
    (CategoryStruct.comp (Y :=
      (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k))))
            (pullback (D.f j i) (D.f j k)).presheaf).obj
        (op ((Opens.map (D.t j i).base).1 ((Opens.map (D.f i j).base).obj U))))
      (NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U) :
        (D.U i).presheaf.obj (op U) ⟶
          (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)))
                (pullback (D.f j i) (D.f j k)).presheaf).obj
            (op U))
      (invApp (pullback.snd (D.f j i) (D.f j k))
          ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k))) :
                Opens (↑(↑(D.V (j, i)) : TopCat) : Type v) ⥤
                  Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
            ((Opens.map (D.t j i).base).1 ((Opens.map (D.f i j).base).obj U))) ≫
        (D.V (j, k)).presheaf.map (eqToHom ⋯)))
    (CategoryStruct.comp (Y :=
      (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)))
            (pullback (D.f j i) (D.f j k)).presheaf).obj
        (op U))
      (NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U))
      (invApp (pullback.snd (D.f j i) (D.f j k))
          ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
            U) ≫
        (D.V (j, k)).presheaf.map (eqToHom ⋯)))

Tactic:
congr
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j k : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
X' : C
f' :
  (Functor.obj (TopCat.Presheaf.pushforward C (D.f j k).base) (D.V (j, k)).presheaf).obj
      (op
        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))) ⟶
    X'
⊢ D.opensImagePreimageMap i j U ≫
      (D.f j k).c.app
          (op
            (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
              (Opens.map (Hom.base (D.ι j)) :
                Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
              (Functor.obj
                  (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                  U :
                Opens (↑(↑D.glued : TopCat) : Type v)))) ≫
        f' =
    NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U) ≫
      invApp (pullback.snd (D.f j i) (D.f j k))
          (unop
            (op
              ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤
                      Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
                (unop (op U))))) ≫
        (D.V (j, k)).presheaf.map (eqToHom ⋯) ≫ f'

Tactic:
simpa only [Category.assoc] using congr_arg (· ≫ f') (opensImagePreimageMap_app D i j k U)

Premises used: CategoryTheory.Category.assoc, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap_app, congr_arg, CategoryTheory.CategoryStruct.comp
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j : WalkingMultispan (MultispanShape.prod D.J)
left : (j k : D.J) → (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (j, k)))
right :
  (j : (MultispanShape.prod D.J).R) →
    (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.right j))
⊢ (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j)

Tactic:
rcases j with (⟨j, k⟩ | j)
-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
left :
  (Opens.map (D.f j k).base).op.obj
      (op
        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))) =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))
⊢ (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (j, k)))

Tactic:
refine
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom ?_)

Premises used: CategoryTheory.eqToHom, CategoryTheory.GlueData.f, Prod.mk, CategoryTheory.Functor.map, CategoryTheory.GlueData.V, AlgebraicGeometry.PresheafedSpace.presheaf, AlgebraicGeometry.PresheafedSpace.Hom.c, AlgebraicGeometry.PresheafedSpace.GlueData.opensImagePreimageMap, CategoryTheory.NatTrans.app, AlgebraicGeometry.PresheafedSpace.GlueData.toGlueData