-- Module: Mathlib.Geometry.Manifold.Sheaf.Smooth
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
EM : Type u_2
inst_1 : NormedAddCommGroup EM
inst_2 : NormedSpace 𝕜 EM
HM : Type u_3
inst_3 : TopologicalSpace HM
IM : ModelWithCorners 𝕜 EM HM
E : Type u_4
inst_4 : NormedAddCommGroup E
inst_5 : NormedSpace 𝕜 E
H : Type u_5
inst_6 : TopologicalSpace H
I : ModelWithCorners 𝕜 E H
H' : Type u_6
inst_7 : TopologicalSpace H'
I' : ModelWithCorners 𝕜 E H'
M : Type u
inst_8 : TopologicalSpace M
inst_9 : ChartedSpace HM M
N G A A' R : Type u
inst_10 : TopologicalSpace N
inst_11 : ChartedSpace H N
inst_12 : TopologicalSpace G
inst_13 : ChartedSpace H G
inst_14 : TopologicalSpace A
inst_15 : ChartedSpace H A
inst_16 : TopologicalSpace A'
inst_17 : ChartedSpace H' A'
inst_18 : TopologicalSpace R
inst_19 : ChartedSpace H R
EP : Type u_7
inst_20 : NormedAddCommGroup EP
inst_21 : NormedSpace 𝕜 EP
HP : Type u_8
inst_22 : TopologicalSpace HP
IP : ModelWithCorners 𝕜 EP HP
P : Type u
inst_23 : TopologicalSpace P
inst_24 : ChartedSpace HP P
inst_25 : CommRing R
inst_26 : ContMDiffRing I ∞ R
x : (↑(TopCat.of M) : Type u)
refine_2 refine_2₁ refine_2₂ refine_2₃ :
  ∀ ⦃X Y : (OpenNhds x)ᵒᵖ⦄ (f : X ⟶ Y),
    Eq (α :=
      (Functor.obj
              (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
                (OpenNhds.inclusion x).op)
              (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).obj
          X ⟶
        Functor.obj (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) Y)
      ((Functor.obj
              (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
                (OpenNhds.inclusion x).op)
              (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).map
          f ≫
        (fun (U : (OpenNhds x)ᵒᵖ) ↦
            (evalAt IM I M R x (unop U) :
              Functor.obj (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))
                  (op (↑(unop U) : Opens (↑(TopCat.of M) : Type u))) ⟶
                CommRingCat.of R))
          Y)
      ((fun (U : (OpenNhds x)ᵒᵖ) ↦
            (evalAt IM I M R x (unop U) :
              Functor.obj (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))
                  (op (↑(unop U) : Opens (↑(TopCat.of M) : Type u))) ⟶
                CommRingCat.of R))
          X ≫
        CategoryTheory.Functor.map (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) f)
⊢ ∀ ⦃X Y : (OpenNhds x)ᵒᵖ⦄ (f : X ⟶ Y),
    Eq (α :=
      (Functor.obj
              (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
                (OpenNhds.inclusion x).op)
              (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).obj
          X ⟶
        Functor.obj (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) Y)
      ((Functor.obj
              (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑(TopCat.of M) : Type u))ᵒᵖ CommRingCat)
                (OpenNhds.inclusion x).op)
              (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))).map
          f ≫
        (fun (U : (OpenNhds x)ᵒᵖ) ↦
            (evalAt IM I M R x (unop U) :
              Functor.obj (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))
                  (op (↑(unop U) : Opens (↑(TopCat.of M) : Type u))) ⟶
                CommRingCat.of R))
          Y)
      ((fun (U : (OpenNhds x)ᵒᵖ) ↦
            (evalAt IM I M R x (unop U) :
              Functor.obj (TopCat.Sheaf.presheaf (smoothSheafCommRing IM I M R))
                  (op (↑(unop U) : Opens (↑(TopCat.of M) : Type u))) ⟶
                CommRingCat.of R))
          X ≫
        CategoryTheory.Functor.map (Functor.obj (Functor.const (OpenNhds x)ᵒᵖ) (CommRingCat.of R)) f)

Tactic:
cat_disch