-- Module: Mathlib.Algebra.Category.ModuleCat.Adjunctions
Goal:
R : Type u
inst : CommRing R
X Y Z : Type u
z : Z
x : X
y : Y
h :
  Eq (α := (↑(Functor.obj (free R) (tensorObj (𝒞 := types) X (tensorObj (𝒞 := types) Y Z))) : Type u))
    ((Hom.hom (CategoryTheory.Functor.map (free R) (Iso.hom (α_ X Y Z))) :
          (a : (↑((free R).obj ((X ⊗ Y) ⊗ Z)) : Type u)) → (↑((free R).obj (X ⊗ Y ⊗ Z)) : Type u))
        ((Hom.hom (Iso.hom (μIso R (tensorObj (𝒞 := types) X Y) Z)) :
            (a : (↑((free R).obj (X ⊗ Y) ⊗ (free R).obj Z) : Type u)) → (↑((free R).obj ((X ⊗ Y) ⊗ Z)) : Type u))
          ((Hom.hom (Iso.hom (μIso R X Y) ▷ Functor.obj (free R) Z) :
              (a : (↑(((free R).obj X ⊗ (free R).obj Y) ⊗ (free R).obj Z) : Type u)) →
                (↑((free R).obj (X ⊗ Y) ⊗ (free R).obj Z) : Type u))
            ((Hom.hom (Iso.inv (μIso R X Y) ▷ Functor.obj (free R) Z) :
                (a : (↑((free R).obj (X ⊗ Y) ⊗ (free R).obj Z) : Type u)) →
                  (↑(((free R).obj X ⊗ (free R).obj Y) ⊗ (free R).obj Z) : Type u))
              ((Hom.hom (Iso.inv (μIso R (tensorObj (𝒞 := types) X Y) Z)) :
                  (a : (↑((free R).obj ((X ⊗ Y) ⊗ Z)) : Type u)) → (↑((free R).obj (X ⊗ Y) ⊗ (free R).obj Z) : Type u))
                (freeMk ((x, y), z)))))) :
      (↑(Functor.obj (free R) (tensorObj (𝒞 := types) X (tensorObj (𝒞 := types) Y Z))) : Type u))
    ((Hom.hom (Iso.hom (μIso R X (tensorObj (𝒞 := types) Y Z))) :
          (a : (↑((free R).obj X ⊗ (free R).obj (Y ⊗ Z)) : Type u)) → (↑((free R).obj (X ⊗ Y ⊗ Z)) : Type u))
        ((Hom.hom (Functor.obj (free R) X ◁ Iso.hom (μIso R Y Z)) :
            (a : (↑((free R).obj X ⊗ (free R).obj Y ⊗ (free R).obj Z) : Type u)) →
              (↑((free R).obj X ⊗ (free R).obj (Y ⊗ Z)) : Type u))
          ((Hom.hom (Iso.hom (α_ (Functor.obj (free R) X) (Functor.obj (free R) Y) (Functor.obj (free R) Z))) :
              (a : (↑(((free R).obj X ⊗ (free R).obj Y) ⊗ (free R).obj Z) : Type u)) →
                (↑((free R).obj X ⊗ (free R).obj Y ⊗ (free R).obj Z) : Type u))
            ((Hom.hom (Iso.inv (μIso R X Y) ▷ Functor.obj (free R) Z) :
                (a : (↑((free R).obj (X ⊗ Y) ⊗ (free R).obj Z) : Type u)) →
                  (↑(((free R).obj X ⊗ (free R).obj Y) ⊗ (free R).obj Z) : Type u))
              ((Hom.hom (Iso.inv (μIso R (tensorObj (𝒞 := types) X Y) Z)) :
                  (a : (↑((free R).obj ((X ⊗ Y) ⊗ Z)) : Type u)) → (↑((free R).obj (X ⊗ Y) ⊗ (free R).obj Z) : Type u))
                (freeMk ((x, y), z)))))) :
      (↑(Functor.obj (free R) (tensorObj (𝒞 := types) X (tensorObj (𝒞 := types) Y Z))) : Type u))
⊢ Eq (α := (↑(Functor.obj (free R) (tensorObj (𝒞 := types) X (tensorObj (𝒞 := types) Y Z))) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC := carrier)
            (Iso.inv (μIso R (tensorObj (𝒞 := types) X Y) Z) ≫
              Iso.inv (μIso R X Y) ▷ Functor.obj (free R) Z ≫
                Iso.hom (μIso R X Y) ▷ Functor.obj (free R) Z ≫
                  Iso.hom (μIso R (tensorObj (𝒞 := types) X Y) Z) ≫
                    CategoryTheory.Functor.map (free R) (Iso.hom (α_ X Y Z))) :
          (a : (↑((free R).obj ((X ⊗ Y) ⊗ Z)) : Type u)) → (↑((free R).obj (X ⊗ Y ⊗ Z)) : Type u))
        (freeMk ((x, y), z)) :
      (↑(Functor.obj (free R) (tensorObj (𝒞 := types) X (tensorObj (𝒞 := types) Y Z))) : Type u))
    ((ConcreteCategory.hom (FC := fun (x1 x2 : ModuleCat R) ↦ (↑x1 : Type u) →ₗ[R] (↑x2 : Type u)) (CC := carrier)
            (Iso.inv (μIso R (tensorObj (𝒞 := types) X Y) Z) ≫
              Iso.inv (μIso R X Y) ▷ Functor.obj (free R) Z ≫
                Iso.hom (α_ (Functor.obj (free R) X) (Functor.obj (free R) Y) (Functor.obj (free R) Z)) ≫
                  Functor.obj (free R) X ◁ Iso.hom (μIso R Y Z) ≫ Iso.hom (μIso R X (tensorObj (𝒞 := types) Y Z))) :
          (a : (↑((free R).obj ((X ⊗ Y) ⊗ Z)) : Type u)) → (↑((free R).obj (X ⊗ Y ⊗ Z)) : Type u))
        (freeMk ((x, y), z)) :
      (↑(Functor.obj (free R) (tensorObj (𝒞 := types) X (tensorObj (𝒞 := types) Y Z))) : Type u))

Tactic:
dsimp