-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
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.U j).presheaf.obj
      (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.right ((MultispanShape.prod D.J).snd (i, j))))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens
                (↑(↑(D.diagram.multispan.obj
                          (unop (op (WalkingMultispan.right ((MultispanShape.prod D.J).snd (i, 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)))))
    ((D.t i j ≫ D.f j i).c.app
      (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.right ((MultispanShape.prod D.J).snd (i, j))))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens
                (↑(↑(D.diagram.multispan.obj
                          (unop (op (WalkingMultispan.right ((MultispanShape.prod D.J).snd (i, 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)))))
have inst : Mono (X :=
  (D.U j).presheaf.obj
    (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.right ((MultispanShape.prod D.J).snd (i, j))))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens
              (↑(↑(D.diagram.multispan.obj
                        (unop (op (WalkingMultispan.right ((MultispanShape.prod D.J).snd (i, j)))))) :
                    TopCat) :
                Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))))
  ((D.f j i).c.app
      (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.right ((MultispanShape.prod D.J).snd (i, j))))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens
                (↑(↑(D.diagram.multispan.obj
                          (unop (op (WalkingMultispan.right ((MultispanShape.prod D.J).snd (i, j)))))) :
                      TopCat) :
                  Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))) ≫
    (D.t i j).c.app
      (op
        ((Opens.map (D.f j i).base).obj
          (unop
            (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.right ((MultispanShape.prod D.J).snd (i, j))))))) :
                  Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
                    Opens
                      (↑(↑(D.diagram.multispan.obj
                                (unop (op (WalkingMultispan.right ((MultispanShape.prod D.J).snd (i, j)))))) :
                            TopCat) :
                        Type v))
                (Functor.obj
                    (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                      Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                    U :
                  Opens (↑(↑D.glued : TopCat) : Type v))))))))
Tactic: rw [comp_c_app]