/-- Converting a bundled algebra to a monoid object in `ModuleCat R`.
-/
@[simps]
def inverseObj (A : AlgebraCat.{u} R) : Mon_ (ModuleCat.{u} R) where
  X := ModuleCat.of R A
  one := Algebra.linearMap R A
  mul := LinearMap.mul' R A
  one_mul := by
    -- Porting note : `ext` did not pick up `TensorProduct.ext`
    refine TensorProduct.ext <| LinearMap.ext_ring <| LinearMap.ext fun x => ?_
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [compr₂_apply, compr₂_apply, CategoryTheory.comp_apply]
    -- Porting note : this `dsimp` does nothing
    -- dsimp [AlgebraCat.id_apply, TensorProduct.mk_apply, Algebra.linearMap_apply,
    --   LinearMap.compr₂_apply, Function.comp_apply, RingHom.map_one,
    --   ModuleCat.MonoidalCategory.hom_apply, AlgebraCat.coe_comp,
    --   ModuleCat.MonoidalCategory.leftUnitor_hom_apply]
    -- Porting note : because `dsimp` is not effective, `rw` needs to be changed to `erw`
    erw [LinearMap.mul'_apply, MonoidalCategory.leftUnitor_hom_apply, ← Algebra.smul_def]
    rw [id_apply]
  mul_one := by
    -- Porting note : `ext` did not pick up `TensorProduct.ext`
    refine TensorProduct.ext <| LinearMap.ext fun x => LinearMap.ext_ring ?_
    -- Porting note : this `dsimp` does nothing
    -- dsimp only [AlgebraCat.id_apply, TensorProduct.mk_apply, Algebra.linearMap_apply,
    --   LinearMap.compr₂_apply, Function.comp_apply, ModuleCat.MonoidalCategory.hom_apply,
    --   AlgebraCat.coe_comp]
    -- Porting note : because `dsimp` is not effective, `rw` needs to be changed to `erw`
    erw [compr₂_apply, compr₂_apply]
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [CategoryTheory.comp_apply]
    erw [LinearMap.mul'_apply, ModuleCat.MonoidalCategory.rightUnitor_hom_apply, ← Algebra.commutes,
      ← Algebra.smul_def]
    rw [id_apply]
  mul_assoc := by
    -- Porting note : `ext` did not pick up `TensorProduct.ext`
    refine TensorProduct.ext <| TensorProduct.ext <| LinearMap.ext fun x => LinearMap.ext fun y =>
      LinearMap.ext fun z => ?_
    -- Porting note : this `dsimp` does nothing
    -- dsimp only [AlgebraCat.id_apply, TensorProduct.mk_apply, LinearMap.compr₂_apply,
    --   Function.comp_apply, ModuleCat.MonoidalCategory.hom_apply, AlgebraCat.coe_comp,
    --   MonoidalCategory.associator_hom_apply]
    -- Porting note : because `dsimp` is not effective, `rw` needs to be changed to `erw`
    rw [compr₂_apply, compr₂_apply, compr₂_apply, compr₂_apply]
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [CategoryTheory.comp_apply,
      CategoryTheory.comp_apply, CategoryTheory.comp_apply]
    

Proof state:
R : Type u
inst✝ : CommRing R
A : AlgebraCat R
x y z : ↑(of R ↑A)
⊢ (mul' R ↑A)
      ((CategoryTheory.MonoidalCategory.tensorHom (mul' R ↑A) (𝟙 (of R ↑A)))
        (((TensorProduct.mk R ↑(CategoryTheory.MonoidalCategory.tensorObj (of R ↑A) (of R ↑A)) ↑(of R ↑A))
            (((TensorProduct.mk R ↑(of R ↑A) ↑(of R ↑A)) x) y))
          z)) =
    (mul' R ↑A)
      ((CategoryTheory.MonoidalCategory.tensorHom (𝟙 (of R ↑A)) (mul' R ↑A))
        ((CategoryTheory.MonoidalCategory.associator (of R ↑A) (of R ↑A) (of R ↑A)).hom
          (((TensorProduct.mk R ↑(CategoryTheory.MonoidalCategory.tensorObj (of R ↑A) (of R ↑A)) ↑(of R ↑A))
              (((TensorProduct.mk R ↑(of R ↑A) ↑(of R ↑A)) x) y))
            z)))

Next tactic:
erw [LinearMap.mul'_apply, LinearMap.mul'_apply]