set_option maxHeartbeats 500000 in
/-- The category of internal monoid objects in `ModuleCat R`
is equivalent to the category of "native" bundled `R`-algebras.
-/
def monModuleEquivalenceAlgebra : Mon_ (ModuleCat.{u} R) ≌ AlgebraCat R where
  functor := functor
  inverse := inverse
  unitIso :=
    NatIso.ofComponents
      (fun A =>
        { hom :=
            { hom :=
                { toFun := _root_.id
                  map_add' := fun x y => rfl
                  map_smul' := fun r a => rfl }
              mul_hom := by
                -- Porting note : `ext` did not pick up `TensorProduct.ext`
                refine TensorProduct.ext ?_
                dsimp at *
                rfl
              one_hom := by ext; rfl }
          inv :=
            { hom :=
                { toFun := _root_.id
                  map_add' := fun x y => rfl
                  map_smul' := fun r a => rfl }
              mul_hom := by
                -- Porting note : `ext` did not pick up `TensorProduct.ext`
                

Proof state:
R : Type u
inst✝ : CommRing R
A : Mon_ (ModuleCat R)
⊢ ((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).mul ≫
      {
        toAddHom :=
          { toFun := _root_.id,
            map_add' :=
              (_ :
                ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                  _root_.id (x + y) = _root_.id (x + y)) },
        map_smul' :=
          (_ :
            ∀ (r : R) (a : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
              AddHom.toFun
                  { toFun := _root_.id,
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                          _root_.id (x + y) = _root_.id (x + y)) }
                  (r • a) =
                AddHom.toFun
                  { toFun := _root_.id,
                    map_add' :=
                      (_ :
                        ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                          _root_.id (x + y) = _root_.id (x + y)) }
                  (r • a)) } =
    CategoryTheory.MonoidalCategory.tensorHom
        {
          toAddHom :=
            { toFun := _root_.id,
              map_add' :=
                (_ :
                  ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                    _root_.id (x + y) = _root_.id (x + y)) },
          map_smul' :=
            (_ :
              ∀ (r : R) (a : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                AddHom.toFun
                    { toFun := _root_.id,
                      map_add' :=
                        (_ :
                          ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                            _root_.id (x + y) = _root_.id (x + y)) }
                    (r • a) =
                  AddHom.toFun
                    { toFun := _root_.id,
                      map_add' :=
                        (_ :
                          ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                            _root_.id (x + y) = _root_.id (x + y)) }
                    (r • a)) }
        {
          toAddHom :=
            { toFun := _root_.id,
              map_add' :=
                (_ :
                  ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                    _root_.id (x + y) = _root_.id (x + y)) },
          map_smul' :=
            (_ :
              ∀ (r : R) (a : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                AddHom.toFun
                    { toFun := _root_.id,
                      map_add' :=
                        (_ :
                          ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                            _root_.id (x + y) = _root_.id (x + y)) }
                    (r • a) =
                  AddHom.toFun
                    { toFun := _root_.id,
                      map_add' :=
                        (_ :
                          ∀ (x y : ↑((functor ⋙ MonModuleEquivalenceAlgebra.inverse).obj A).X),
                            _root_.id (x + y) = _root_.id (x + y)) }
                    (r • a)) } ≫
      ((𝟭 (Mon_ (ModuleCat R))).obj A).mul

Next tactic:
refine TensorProduct.ext ?_