/-- Verify that the `pushout_cocone` is indeed the colimit. -/
def pushoutCoconeIsColimit : Limits.IsColimit (pushoutCocone f g) :=
  Limits.PushoutCocone.isColimitAux' _ fun s => by
    letI := RingHom.toAlgebra f
    letI := RingHom.toAlgebra g
    letI := RingHom.toAlgebra (f ≫ s.inl)
    let f' : A →ₐ[R] s.pt :=
      { s.inl with
        commutes' := fun r => rfl }
    let g' : B →ₐ[R] s.pt :=
      { s.inr with
        commutes' := fun r => by
          change (g ≫ s.inr) r = (f ≫ s.inl) r
          congr 1
          exact
            (s.ι.naturality Limits.WalkingSpan.Hom.snd).trans
              (s.ι.naturality Limits.WalkingSpan.Hom.fst).symm }
    -- Porting note : Lean has forget why `A ⊗[R] B` makes sense
    letI : Algebra R A := f.toAlgebra
    letI : Algebra R B := g.toAlgebra
    letI : Algebra R (pushoutCocone f g).pt := show Algebra R (A ⊗[R] B) by infer_instance
    -- The factor map is a ⊗ b ↦ f(a) * g(b).
    use AlgHom.toRingHom (Algebra.TensorProduct.productMap f' g')
    

Proof state:
case property
R A B : CommRingCat
f : R ⟶ A
g : R ⟶ B
s : PushoutCocone f g
this✝⁴ : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝³ : Algebra ↑R ↑B := RingHom.toAlgebra g
this✝² : Algebra ↑R ↑s.pt := RingHom.toAlgebra (f ≫ PushoutCocone.inl s)
f' : ↑A →ₐ[↑R] ↑s.pt :=
  let src := PushoutCocone.inl s;
  {
    toRingHom :=
      { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
        map_add' :=
          (_ : ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
    commutes' :=
      (_ :
        ∀ (r : ↑R),
          OneHom.toFun
              (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })
              ((algebraMap ↑R ↑A) r) =
            OneHom.toFun
              (↑↑{ toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑A), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) })
              ((algebraMap ↑R ↑A) r)) }
g' : ↑B →ₐ[↑R] ↑s.pt :=
  let src := PushoutCocone.inr s;
  {
    toRingHom :=
      { toMonoidHom := ↑src, map_zero' := (_ : OneHom.toFun (↑↑src) 0 = 0),
        map_add' :=
          (_ : ∀ (x y : ↑B), OneHom.toFun (↑↑src) (x + y) = OneHom.toFun (↑↑src) x + OneHom.toFun (↑↑src) y) },
    commutes' := (_ : ∀ (r : ↑R), (g ≫ PushoutCocone.inr s) r = (f ≫ PushoutCocone.inl s) r) }
this✝¹ : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝ : Algebra ↑R ↑B := RingHom.toAlgebra g
this : Algebra ↑R ↑(pushoutCocone f g).pt :=
  let_fun this := inferInstance;
  this
⊢ PushoutCocone.inl (pushoutCocone f g) ≫ ↑(Algebra.TensorProduct.productMap f' g') = PushoutCocone.inl s ∧
    PushoutCocone.inr (pushoutCocone f g) ≫ ↑(Algebra.TensorProduct.productMap f' g') = PushoutCocone.inr s ∧
      ∀ {m : (pushoutCocone f g).pt ⟶ s.pt},
        PushoutCocone.inl (pushoutCocone f g) ≫ m = PushoutCocone.inl s →
          PushoutCocone.inr (pushoutCocone f g) ≫ m = PushoutCocone.inr s →
            m = ↑(Algebra.TensorProduct.productMap f' g')

Next tactic:
simp only [pushoutCocone_inl, pushoutCocone_inr]