-- 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 ((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 ⋯)))
⊢ 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 ⋯ ≫ Quiver.Hom.op ((Opens.map (D.f j k).base).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:
rw [eqToHom_map (Opens.map _), eqToHom_op, eqToHom_trans]

Premises used: CategoryTheory.eqToHom_map, CategoryTheory.eqToHom_op, TopologicalSpace.Opens.map, CategoryTheory.eqToHom_trans