-- Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : LocallySmall.{w, v, u} C
inst_2 : IsCofiltered.{v, u} C
inst_3 : InitiallySmall.{w, v, u} C
R : CategoryTheory.Functor.{v, w, u, w + 1} Cᵒᵖ RingCat
cR : Cocone R
hcR : IsColimit cR
M : PresheafOfModules R
cM : Cocone M.presheaf
hcM : IsColimit cM
M' : PresheafOfModules R
cM' : Cocone M'.presheaf
hcM' : IsColimit cM'
r : (↑cR.pt : Type w)
m₁ : ModuleColimit hcR hcM
m₂ : ModuleColimit hcR hcM'
U : Cᵒᵖ
a : (R ⋙ forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC := RingCat.carrier)).obj U
x₁ :
  (M.presheaf ⋙
        forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
          AddCommGrpCat.carrier)).obj
    U
x₂ :
  (M'.presheaf ⋙
        forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
          AddCommGrpCat.carrier)).obj
    U
h :
  (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (Cocone.ι
              (Cocone.tensor
                (Functor.mapCocone
                  (forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
                    RingCat.carrier))
                  cR)
                (Cocone.tensor
                  (Functor.mapCocone
                    (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w))
                      (CC := AddCommGrpCat.carrier))
                    cM)
                  (Functor.mapCocone
                    (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w))
                      (CC := AddCommGrpCat.carrier))
                    cM'))))
            U) :
        (a :
            (fun (X : Type w) ↦ X)
              ((R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat ⊗ M'.presheaf ⋙ forget AddCommGrpCat).obj U)) →
          (fun (X : Type w) ↦ X)
            (((Functor.const Cᵒᵖ).obj
                  (((forget RingCat).mapCocone cR).tensor
                      (((forget AddCommGrpCat).mapCocone cM).tensor ((forget AddCommGrpCat).mapCocone cM'))).pt).obj
              U))
      (a, x₁, x₂) =
    (r, m₁, m₂)
⊢ ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app
                (Cocone.ι
                  (Cocone.tensor
                    (Functor.mapCocone
                      (forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
                        RingCat.carrier))
                      cR)
                    (Cocone.tensor
                      (Functor.mapCocone
                        (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w))
                          (CC := AddCommGrpCat.carrier))
                        cM)
                      (Functor.mapCocone
                        (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w))
                          (CC := AddCommGrpCat.carrier))
                        cM'))))
                U) :
            (a :
                (fun (X : Type w) ↦ X)
                  ((R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat ⊗ M'.presheaf ⋙ forget AddCommGrpCat).obj
                    U)) →
              (fun (X : Type w) ↦ X)
                (((Functor.const Cᵒᵖ).obj
                      (((forget RingCat).mapCocone cR).tensor
                          (((forget AddCommGrpCat).mapCocone cM).tensor ((forget AddCommGrpCat).mapCocone cM'))).pt).obj
                  U))
          (a, x₁, x₂)).1 =
      (r, m₁, m₂).1 ∧
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app
                    (Cocone.ι
                      (Cocone.tensor
                        (Functor.mapCocone
                          (forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
                            RingCat.carrier))
                          cR)
                        (Cocone.tensor
                          (Functor.mapCocone
                            (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦
                              (↑x1 : Type w) →+ (↑x2 : Type w)) (CC := AddCommGrpCat.carrier))
                            cM)
                          (Functor.mapCocone
                            (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦
                              (↑x1 : Type w) →+ (↑x2 : Type w)) (CC := AddCommGrpCat.carrier))
                            cM'))))
                    U) :
                (a :
                    (fun (X : Type w) ↦ X)
                      ((R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat ⊗ M'.presheaf ⋙ forget AddCommGrpCat).obj
                        U)) →
                  (fun (X : Type w) ↦ X)
                    (((Functor.const Cᵒᵖ).obj
                          (((forget RingCat).mapCocone cR).tensor
                              (((forget AddCommGrpCat).mapCocone cM).tensor
                                ((forget AddCommGrpCat).mapCocone cM'))).pt).obj
                      U))
              (a, x₁, x₂)).2.1 =
        (r, m₁, m₂).2.1 ∧
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
                  (NatTrans.app
                    (Cocone.ι
                      (Cocone.tensor
                        (Functor.mapCocone
                          (forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
                            RingCat.carrier))
                          cR)
                        (Cocone.tensor
                          (Functor.mapCocone
                            (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦
                              (↑x1 : Type w) →+ (↑x2 : Type w)) (CC := AddCommGrpCat.carrier))
                            cM)
                          (Functor.mapCocone
                            (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦
                              (↑x1 : Type w) →+ (↑x2 : Type w)) (CC := AddCommGrpCat.carrier))
                            cM'))))
                    U) :
                (a :
                    (fun (X : Type w) ↦ X)
                      ((R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat ⊗ M'.presheaf ⋙ forget AddCommGrpCat).obj
                        U)) →
                  (fun (X : Type w) ↦ X)
                    (((Functor.const Cᵒᵖ).obj
                          (((forget RingCat).mapCocone cR).tensor
                              (((forget AddCommGrpCat).mapCocone cM).tensor
                                ((forget AddCommGrpCat).mapCocone cM'))).pt).obj
                      U))
              (a, x₁, x₂)).2.2 =
        (r, m₁, m₂).2.2

Tactic:
rw [Prod.ext_iff, Prod.ext_iff] at h

Premises used: Prod.ext_iff