-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst_1 : Category.{v, u} C
D : GlueData.{v, u} C
inst_2 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
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)) ⋯ :
                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)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : 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.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)))) ≫
      (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)) ⋯ :
                        Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                      U :
                    Opens (↑(↑D.glued : TopCat) : Type v))))))))
have inst₁ : Mono (X :=
  (Functor.obj (TopCat.Presheaf.pushforward C (D.f j i).base) (D.V (j, i)).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.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: apply +allowSynthFailures mono_comp