-- Lean 4 proof step from Mathlib.Algebra.Category.ModuleCat.Presheaf.ColimitFunctor (goal)
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)).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
⊢ ∃ (U : Cᵒᵖ) (a : (↑(R.obj U) : Type w)) (x : (↑(M.obj U) : Type w)),
    (ιR cR : (a : (↑(R.obj U) : Type w)) → (↑cR.pt : Type w)) a = r ∧
      (ιM : (a : (↑(M.obj U) : Type w)) → ModuleColimit hcR hcM) x = m
have obtain : ∃ (U_1 : Cᵒᵖ) (a_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)) a_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)
                      (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 :
        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)
                      (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 :
        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))
Tactic: obtain ⟨rfl, rfl⟩ := h
Cleared hypotheses: r, m, h