-- Module: Mathlib.Algebra.Category.ModuleCat.Stalk
Goal:
C : Type u_1
inst : SmallCategory C
inst_1 : IsFiltered C
R : C ⥤ RingCat
M : C ⥤ Ab
inst_2 : (i : C) → Module (↑(R.obj i) : Type (max u_1 u_2)) (↑(M.obj i) : Type (max u_1 u_3))
H :
  ∀ {i j : C} (f : i ⟶ j) (r : (↑(R.obj i) : Type (max u_1 u_2))) (m : (↑(M.obj i) : Type (max u_1 u_3))),
    (ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat) ↦ (↑x1 : Type (max u_1 u_3)) →+ (↑x2 : Type (max u_1 u_3)))
            (CC := AddCommGrpCat.carrier) (M.map f) :
          (a : (↑(M.obj i) : Type (max u_1 u_3))) → (↑(M.obj j) : Type (max u_1 u_3)))
        (r • m) =
      HSMul.hSMul (α := (↑(R.obj j) : Type (max u_1 u_2))) (β := (↑(M.obj j) : Type (max u_1 u_3)))
        ((ConcreteCategory.hom (FC := fun (R S : RingCat) ↦ (↑R : Type (max u_1 u_2)) →+* (↑S : Type (max u_1 u_2)))
                (CC := RingCat.carrier) (R.map f) :
              (a : (↑(R.obj i) : Type (max u_1 u_2))) → (↑(R.obj j) : Type (max u_1 u_2)))
            r :
          (↑(R.obj j) : Type (max u_1 u_2)))
        ((ConcreteCategory.hom (FC := fun (x1 x2 : AddCommGrpCat) ↦
                (↑x1 : Type (max u_1 u_3)) →+ (↑x2 : Type (max u_1 u_3))) (CC := AddCommGrpCat.carrier) (M.map f) :
              (a : (↑(M.obj i) : Type (max u_1 u_3))) → (↑(M.obj j) : Type (max u_1 u_3)))
            m :
          (↑(M.obj j) : Type (max u_1 u_3)))
r : (↑(RingCat.FilteredColimits.colimit.{u_1, u_2} R) : Type (max u_1 u_2))
U : C
a :
  (((R ⋙
            forget₂ RingCat (FC := fun (R S : RingCat) ↦ (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CC :=
              RingCat.carrier) SemiRingCat (FD := fun (R S : SemiRingCat) ↦
              (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CD := SemiRingCat.carrier)) ⋙
          forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1)))
            (CC := SemiRingCat.carrier) MonCat (FD := fun (x1 x2 : MonCat) ↦
            (↑x1 : Type (max u_2 u_1)) →* (↑x2 : Type (max u_2 u_1))) (CD := MonCat.carrier)) ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u_2 u_1)) →* (↑x2 : Type (max u_2 u_1))) (CC :=
          MonCat.carrier)).obj
    U
⊢ HSMul.hSMul (β := (↑(AddCommGrpCat.FilteredColimits.colimit.{u_1, u_3} M) : Type (max u_1 u_3)))
      (Quot.mk
        (((R ⋙
                forget₂ RingCat (FC := fun (R S : RingCat) ↦ (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1)))
                  (CC := RingCat.carrier) SemiRingCat (FD := fun (R S : SemiRingCat) ↦
                  (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CD := SemiRingCat.carrier)) ⋙
              forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦
                (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CC := SemiRingCat.carrier) MonCat (FD :=
                fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u_2 u_1)) →* (↑x2 : Type (max u_2 u_1))) (CD :=
                MonCat.carrier)) ⋙
            forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u_2 u_1)) →* (↑x2 : Type (max u_2 u_1))) (CC :=
              MonCat.carrier)).ColimitTypeRel
        (Sigma.mk (β := fun (j : C) ↦
          (((R ⋙
                    forget₂ RingCat (FC := fun (R S : RingCat) ↦
                      (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CC := RingCat.carrier) SemiRingCat
                      (FD := fun (R S : SemiRingCat) ↦ (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CD :=
                      SemiRingCat.carrier)) ⋙
                  forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦
                    (↑R : Type (max u_2 u_1)) →+* (↑S : Type (max u_2 u_1))) (CC := SemiRingCat.carrier) MonCat (FD :=
                    fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u_2 u_1)) →* (↑x2 : Type (max u_2 u_1))) (CD :=
                    MonCat.carrier)) ⋙
                forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u_2 u_1)) →* (↑x2 : Type (max u_2 u_1)))
                  (CC := MonCat.carrier)).obj
            j)
          U a))
      0 =
    0

Tactic:
Functor.ιColimitType_eq_of_map_eq_map _ _ _ (𝟙 _) (rightToMax _ _) ?_

Premises used: CategoryTheory.IsFiltered.rightToMax, CategoryTheory.Functor.ιColimitType_eq_of_map_eq_map, Quot.lift₂, CategoryTheory.Functor.comp, CategoryTheory.CategoryStruct.id, CategoryTheory.Functor.ColimitType, Quot.liftOn₂, CategoryTheory.Limits.colimit.smul, CategoryTheory.Functor.ιColimitType