-- 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)
h :
  Eq (α :=
    (D.U i).presheaf.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 i).c.app (op U) ≫
        (D.V (i, i)).presheaf.map (𝟙 (op ((Opens.map (D.f i i).base).obj U)) ≫ eqToHom ⋯) ≫
          invApp (D.f i i) ((Opens.map (D.t i i).base).1 ((Opens.map (D.f i i).base).obj U)) ≫
            (D.U i).presheaf.map (Quiver.Hom.op (eqToHom ⋯)) :
      (D.U i).presheaf.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.U i).presheaf.map (eqToHom ⋯) :
      (D.U i).presheaf.obj (op U) ⟶
        (D.U i).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
              (Opens.map (Hom.base (Multicoequalizer.π D.diagram i)) :
                Opens (↑(↑(Limits.colimit D.diagram.multispan) : 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)))))
⊢ Eq (α :=
    (D.U i).presheaf.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 i).c.app (op U) ≫
        (D.t i i).c.app (op ((Opens.map (D.f i i).base).obj U)) ≫
          invApp (D.f i i) ((Opens.map (D.t i i).base).1 ((Opens.map (D.f i i).base).obj U)) ≫
            (D.U i).presheaf.map (Quiver.Hom.op (eqToHom ⋯)) :
      (D.U i).presheaf.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.U i).presheaf.map (eqToHom ⋯) :
      (D.U i).presheaf.obj (op U) ⟶
        (D.U i).presheaf.obj
          (op
            (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
              (Opens.map (Hom.base (Multicoequalizer.π D.diagram i)) :
                Opens (↑(↑(Limits.colimit D.diagram.multispan) : 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)))))

Tactic:
rw [congr_app (D.t_id _), id_c_app, ← Functor.map_comp]

Premises used: CategoryTheory.GlueData.t_id, CategoryTheory.Functor.map_comp, AlgebraicGeometry.PresheafedSpace.congr_app, AlgebraicGeometry.PresheafedSpace.id_c_app, AlgebraicGeometry.PresheafedSpace.GlueData.toGlueData