-- 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
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)).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
obtain :
  ∃ (U_1 : Cᵒᵖ) (a₁_1 : (↑(R.obj U_1) : Type w)) (a₂_1 : (↑(R.obj U_1) : Type w)) (x_1 : (↑(M.obj U_1) : Type w)),
    Eq (α := (↑cR.pt : Type w)) ((ιR cR : (a : (↑(R.obj U_1) : Type w)) → (↑cR.pt : Type w)) a₁_1 : (↑cR.pt : Type w))
        (((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 :
          Cocone.pt (F :=
            R ⋙
              forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC := RingCat.carrier))
            (Functor.mapCocone
              (forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC := RingCat.carrier))
              cR)) ∧
      Eq (α := (↑cR.pt : Type w)) ((ιR cR : (a : (↑(R.obj U_1) : Type w)) → (↑cR.pt : Type w)) a₂_1 : (↑cR.pt : Type w))
          (((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 :
            Cocone.pt (F :=
              R ⋙
                forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
                  RingCat.carrier))
              (Functor.mapCocone
                (forget RingCat (FC := fun (R S : RingCat.{w}) ↦ (↑R : Type w) →+* (↑S : Type w)) (CC :=
                  RingCat.carrier))
                cR)) ∧
        Eq (α := ModuleColimit hcR hcM)
          ((ιM : (a : (↑(M.obj U_1) : Type w)) → ModuleColimit hcR hcM) x_1 : ModuleColimit hcR hcM)
          (((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 :
            Cocone.pt (F :=
              M.presheaf ⋙
                forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
                  AddCommGrpCat.carrier))
              (Functor.mapCocone
                (forget AddCommGrpCat (FC := fun (x1 x2 : AddCommGrpCat.{w}) ↦ (↑x1 : Type w) →+ (↑x2 : Type w)) (CC :=
                  AddCommGrpCat.carrier))
                cM))
⊢ ∃ (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

Tactic:
obtain ⟨rfl, rfl, rfl⟩ := h

Premises used: rfl