-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.Closed.Basic (goal)
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : MonoidalCategory.{v, u} C
D : Type u₂
inst_2 : Category.{v₂, u₂} D
inst_3 : MonoidalCategory.{v₂, u₂} D
F : Functor.{v, v₂, u, u₂} C D
G : Functor.{v₂, v, u₂, u} D C
adj : F ⊣ G
inst_4 : F.Monoidal
inst_5 : F.IsEquivalence
inst_6 : MonoidalClosed.{v₂, u₂} D
X Y Z : C
f : tensorObj.{v, u} (𝒞 := inst) X Y ⟶ Z
⊢ Eq (α := Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))
    ((Adjunction.homEquiv (adj.comp (Adjunction.comp (ihom.adjunction (F.obj X)) adj.toEquivalence.symm.toAdjunction)) Y
            Z :
          (a : G.obj (F.obj X ⊗ F.obj Y) ⟶ Z) → Y ⟶ G.obj ((ihom (F.obj X)).obj (F.obj Z)))
        (NatTrans.app (Iso.hom (Iso.compInverseIso (Functor.Monoidal.commTensorLeft F X))) Y ≫ f :
          G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) ⟶ Z) :
      Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))
    (((adj.homEquiv Y (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)) :
            Equiv.{v₂ + 1, v + 1} (F.obj Y ⟶ Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z))
              (Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))) :
          (a : F.obj Y ⟶ (ihom (F.obj X)).obj (F.obj Z)) → Y ⟶ G.obj ((ihom (F.obj X)).obj (F.obj Z)))
        (curry
          (((adj.toEquivalence.symm.toAdjunction.homEquiv (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) Z :
                Equiv.{v + 1, v₂ + 1} (G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) ⟶ Z)
                  (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ⟶ F.obj Z)) :
              (a : G.obj (F.obj X ⊗ F.obj Y) ⟶ Z) → F.obj X ⊗ F.obj Y ⟶ F.obj Z)
            (NatTrans.app (Iso.hom (Iso.compInverseIso (Functor.Monoidal.commTensorLeft F X))) Y ≫ f))) :
      Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))
have rw : Eq.{v + 1} (α := Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))
  (((fun (x x_1 : C) ↦
            (Equiv.trans
                ((fun (x : D) (x_2 : C) ↦
                    (Equiv.trans
                        (adj.toEquivalence.symm.toAdjunction.homEquiv
                          (Functor.obj.{v₂, v₂, u₂, u₂} (tensorLeft (F.obj X)) x) x_2)
                        (Adjunction.homEquiv (ihom.adjunction (F.obj X)) x (F.obj x_2)) :
                      Equiv.{v + 1, v₂ + 1} (G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (tensorLeft (F.obj X)) x) ⟶ x_2)
                        (x ⟶ Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj x_2))))
                  (F.obj x) x_1)
                (adj.homEquiv x ((F ⋙ ihom (F.obj X)).obj x_1)) :
              ((tensorLeft (F.obj X) ⋙ G).obj (F.obj x) ⟶ x_1) ≃ (x ⟶ G.obj ((F ⋙ ihom (F.obj X)).obj x_1))))
          Y Z :
        (a : G.obj (F.obj X ⊗ F.obj Y) ⟶ Z) → Y ⟶ G.obj ((ihom (F.obj X)).obj (F.obj Z)))
      (NatTrans.app (Iso.hom (Iso.compInverseIso (Functor.Monoidal.commTensorLeft F X))) Y ≫ f :
        G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) ⟶ Z) :
    Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))
  (((adj.homEquiv Y (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)) :
          Equiv.{v₂ + 1, v + 1} (F.obj Y ⟶ Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z))
            (Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))) :
        (a : F.obj Y ⟶ (ihom (F.obj X)).obj (F.obj Z)) → Y ⟶ G.obj ((ihom (F.obj X)).obj (F.obj Z)))
      (curry
        (((adj.toEquivalence.symm.toAdjunction.homEquiv (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) Z :
              Equiv.{v + 1, v₂ + 1} (G.obj (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y)) ⟶ Z)
                (tensorObj.{v₂, u₂} (𝒞 := inst_2) (F.obj X) (F.obj Y) ⟶ F.obj Z)) :
            (a : G.obj (F.obj X ⊗ F.obj Y) ⟶ Z) → F.obj X ⊗ F.obj Y ⟶ F.obj Z)
          (NatTrans.app (Iso.hom (Iso.compInverseIso (Functor.Monoidal.commTensorLeft F X))) Y ≫ f))) :
    Y ⟶ G.obj (Functor.obj.{v₂, v₂, u₂, u₂} (ihom (F.obj X)) (F.obj Z)))
Tactic: rw [Adjunction.comp_homEquiv, Adjunction.comp_homEquiv]