/-- 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')
    simp only [pushoutCocone_inl, pushoutCocone_inr]
    constructor
    · ext x
      -- Porting note : Lean can't see through `forget` functor
      letI : Semiring ((forget CommRingCat).obj A) := A.str.toSemiring
      letI : Algebra R ((forget CommRingCat).obj A) := show Algebra R A by infer_instance
      exact Algebra.TensorProduct.productMap_left_apply _ _ x
    constructor
    · ext x
      -- Porting note : Lean can't see through `forget` functor
      letI : Semiring ((forget CommRingCat).obj B) := B.str.toSemiring
      letI : Algebra R ((forget CommRingCat).obj B) := show Algebra R B by infer_instance
      exact Algebra.TensorProduct.productMap_right_apply _ _ x
    intro h eq1 eq2
    let h' : A ⊗[R] B →ₐ[R] s.pt :=
      { h with
        commutes' := fun r => by
          change h (f r ⊗ₜ[R] 1) = s.inl (f r)
          rw [← eq1]
          simp only [pushoutCocone_pt, coe_of, AlgHom.toRingHom_eq_coe]
          rfl }
    suffices h' = Algebra.TensorProduct.productMap f' g' by
      ext x
      change h' x = Algebra.TensorProduct.productMap f' g' x
      

Proof state:
case w
R A B : CommRingCat
f : R ⟶ A
g : R ⟶ B
s : PushoutCocone f g
this✝⁵ : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝⁴ : Algebra ↑R ↑B := RingHom.toAlgebra g
this✝³ : 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✝² : Algebra ↑R ↑A := RingHom.toAlgebra f
this✝¹ : Algebra ↑R ↑B := RingHom.toAlgebra g
this✝ : Algebra ↑R ↑(pushoutCocone f g).pt :=
  let_fun this := inferInstance;
  this
h : (pushoutCocone f g).pt ⟶ s.pt
eq1 : Algebra.TensorProduct.includeLeftRingHom ≫ h = PushoutCocone.inl s
eq2 : ↑Algebra.TensorProduct.includeRight ≫ h = PushoutCocone.inr s
h' : ↑A ⊗[↑R] ↑B →ₐ[↑R] ↑s.pt :=
  {
    toRingHom :=
      { toMonoidHom := ↑h, map_zero' := (_ : OneHom.toFun (↑↑h) 0 = 0),
        map_add' :=
          (_ :
            ∀ (x y : ↑(pushoutCocone f g).pt),
              OneHom.toFun (↑↑h) (x + y) = OneHom.toFun (↑↑h) x + OneHom.toFun (↑↑h) y) },
    commutes' := (_ : ∀ (r : ↑R), h (f r ⊗ₜ[↑R] 1) = (PushoutCocone.inl s) (f r)) }
this : h' = Algebra.TensorProduct.productMap f' g'
x : (forget CommRingCat).obj (pushoutCocone f g).pt
⊢ h' x = (Algebra.TensorProduct.productMap f' g') x

Next tactic:
rw [this]