-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor (hypothesis)
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
r₁ r₂ : (↑cR.pt : Type w)
m : ModuleColimit hcR hcM
U : Cᵒᵖ
a₁ a₂ :
  (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)
                (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 ⊗ R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat).obj U)) →
          (fun (X : Type w) ↦ X)
            (((Functor.const Cᵒᵖ).obj
                  (((forget RingCat).mapCocone cR).tensor
                      (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM))).pt).obj
              U))
      (a₁, a₂, x) =
    (r₁, r₂, m)
⊢ ∃ (U : Cᵒᵖ) (a₁ : (↑(R.obj U) : Type w)) (a₂ : (↑(R.obj U) : Type w)) (x : (↑(M.obj U) : Type w)),
    (ιR cR : (a : (↑(R.obj U) : Type w)) → (↑cR.pt : Type w)) a₁ = r₁ ∧
      (ιR cR : (a : (↑(R.obj U) : Type w)) → (↑cR.pt : Type w)) a₂ = r₂ ∧
        (ιM : (a : (↑(M.obj U) : Type w)) → ModuleColimit hcR hcM) x = m
have h₁ : ((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 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 ⊗ R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat).obj U)) →
            (fun (X : Type w) ↦ X)
              (((Functor.const Cᵒᵖ).obj
                    (((forget RingCat).mapCocone cR).tensor
                        (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM))).pt).obj
                U))
        (a₁, a₂, x)).1 =
    (r₁, r₂, 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)
                      (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 ⊗ R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat).obj U)) →
                (fun (X : Type w) ↦ X)
                  (((Functor.const Cᵒᵖ).obj
                        (((forget RingCat).mapCocone cR).tensor
                            (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM))).pt).obj
                    U))
            (a₁, a₂, x)).2.1 =
      (r₁, r₂, m).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 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 ⊗ R ⋙ forget RingCat ⊗ M.presheaf ⋙ forget AddCommGrpCat).obj U)) →
                (fun (X : Type w) ↦ X)
                  (((Functor.const Cᵒᵖ).obj
                        (((forget RingCat).mapCocone cR).tensor
                            (((forget RingCat).mapCocone cR).tensor ((forget AddCommGrpCat).mapCocone cM))).pt).obj
                    U))
            (a₁, a₂, x)).2.2 =
      (r₁, r₂, m).2.2
Tactic: rw [Prod.ext_iff, Prod.ext_iff] at h
Cleared hypotheses: h