-- 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)
rw :
  Eq (α :=
    limit (D.diagramOverOpen U) ⟶
      (componentwiseDiagram D.diagram.multispan
            (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))).obj
        (op (WalkingMultispan.left (i, j))))
    (D.diagramOverOpenπ U i ≫
        (D.U i).presheaf.map
            (Iso.inv (eqToIso ⋯) ≫
              Quiver.Hom.op (X :=
                Set.image (α := (↑(↑(D.V (i, 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 i).base) :
                    (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(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 i).base) :
                      (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
                    U.carrier))
                (Y := U.carrier)
                (homOfLE (x :=
                  Set.image (α := (↑(↑(D.V (i, 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 i).base) :
                      (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(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 i).base) :
                        (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
                      U.carrier))
                  (y := U.carrier) ⋯)) ≫
          (D.U i).presheaf.map
              ((opensFunctor (D.f i i)).op.map (𝟙 ((Opens.map (D.f i i).base).op.obj (op U)) ≫ eqToHom ⋯)) ≫
            CategoryStruct.comp (Y :=
              (D.U i).presheaf.obj
                (op
                  (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                    (Opens.map (Hom.base (D.ι i)) :
                      Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.map (Quiver.Hom.op (eqToHom ⋯)) :
                (D.U i).presheaf.obj
                    (op
                      ((opensFunctor (D.f i i)).obj
                        ((Opens.map (D.t i i).base).1 (unop ((Opens.map (D.f i i).base).op.obj (op U)))))) ⟶
                  (D.U i).presheaf.obj
                    (op
                      (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                        (Opens.map (Hom.base (D.ι i)) :
                          Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.f i j).c.app
                    (op
                      (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                        (Opens.map (Hom.base (D.ι i)) :
                          Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.V (i, j)).presheaf.map (eqToHom ⋯) :
                (D.U i).presheaf.obj
                    (op
                      (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                        (Opens.map (Hom.base (D.ι i)) :
                          Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.diagramOverOpen U).obj (op (WalkingMultispan.left (i, j)))) :
      limit (D.diagramOverOpen U) ⟶
        (componentwiseDiagram D.diagram.multispan
              (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))).obj
          (op (WalkingMultispan.left (i, j))))
    (limit.π (D.diagramOverOpen U) (op (WalkingMultispan.left (i, j))) :
      limit (D.diagramOverOpen U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (i, j))))
⊢ Eq (α :=
    limit (D.diagramOverOpen U) ⟶
      (componentwiseDiagram D.diagram.multispan
            (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))).obj
        (op (WalkingMultispan.left (i, j))))
    (D.diagramOverOpenπ U i ≫
        D.ιInvAppπEqMap U ≫
          (D.U i).presheaf.map
              (Quiver.Hom.op (X :=
                Set.image (α := (↑(↑(D.V (i, 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 i).base) :
                    (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(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 i).base) :
                      (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
                    U.carrier))
                (Y := U.carrier)
                (homOfLE (x :=
                  Set.image (α := (↑(↑(D.V (i, 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 i).base) :
                      (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(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 i).base) :
                        (a : (↑(↑(D.V (i, i)) : TopCat) : Type v)) → (↑(↑(D.U i) : TopCat) : Type v)) ⁻¹'
                      U.carrier))
                  (y := U.carrier) ⋯)) ≫
            (D.U i).presheaf.map
                ((opensFunctor (D.f i i)).op.map (𝟙 ((Opens.map (D.f i i).base).op.obj (op U)) ≫ eqToHom ⋯)) ≫
              CategoryStruct.comp (Y :=
                (D.U i).presheaf.obj
                  (op
                    (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                      (Opens.map (Hom.base (D.ι i)) :
                        Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.map (Quiver.Hom.op (eqToHom ⋯)) :
                  (D.U i).presheaf.obj
                      (op
                        ((opensFunctor (D.f i i)).obj
                          ((Opens.map (D.t i i).base).1 (unop ((Opens.map (D.f i i).base).op.obj (op U)))))) ⟶
                    (D.U i).presheaf.obj
                      (op
                        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                          (Opens.map (Hom.base (D.ι i)) :
                            Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.f i j).c.app
                      (op
                        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                          (Opens.map (Hom.base (D.ι i)) :
                            Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.V (i, j)).presheaf.map (eqToHom ⋯) :
                  (D.U i).presheaf.obj
                      (op
                        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
                          (Opens.map (Hom.base (D.ι i)) :
                            Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U i) : 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.diagramOverOpen U).obj (op (WalkingMultispan.left (i, j)))) :
      limit (D.diagramOverOpen U) ⟶
        (componentwiseDiagram D.diagram.multispan
              (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))).obj
          (op (WalkingMultispan.left (i, j))))
    (limit.π (D.diagramOverOpen U) (op (WalkingMultispan.left (i, j))) :
      limit (D.diagramOverOpen U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (i, j))))

Tactic:
rw [← Functor.map_comp_assoc]

Premises used: CategoryTheory.Functor.map_comp_assoc