-- 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
r : (↑cR.pt : Type w)
m : 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
x :
  (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)
                (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).obj U)) →
          (fun (X : Type w) ↦ X)
            (((Functor.const Cᵒᵖ).obj
                  (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM)).pt).obj
              U))
      (a, x) =
    (r, m)
⊢ ((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)
                    (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).obj U)) →
              (fun (X : Type w) ↦ X)
                (((Functor.const Cᵒᵖ).obj
                      (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM)).pt).obj
                  U))
          (a, x)).1 =
      (r, m).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)
                    (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).obj U)) →
              (fun (X : Type w) ↦ X)
                (((Functor.const Cᵒᵖ).obj
                      (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM)).pt).obj
                  U))
          (a, x)).2 =
      (r, m).2

Tactic:
rw [Prod.ext_iff] at h

Premises used: Prod.ext_iff