-- Module: Mathlib.CategoryTheory.Monoidal.Closed.Basic
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
rw :
  Eq (α := Y ⟶ ((adj.toEquivalence.symm.inverse ⋙ ihom (F.obj X)) ⋙ G).obj Z)
    ((Adjunction.homEquiv (adj.comp (Adjunction.comp (ihom.adjunction (F.obj X)) adj.toEquivalence.symm.toAdjunction)) Y
            Z :
          (a : (F ⋙ tensorLeft (F.obj X) ⋙ adj.toEquivalence.symm.functor).obj Y ⟶ Z) →
            Y ⟶ ((adj.toEquivalence.symm.inverse ⋙ ihom (F.obj X)) ⋙ G).obj Z)
        (NatTrans.app (Iso.hom (Iso.compInverseIso (Functor.Monoidal.commTensorLeft F X))) Y ≫ f :
          (F ⋙ tensorLeft (F.obj X) ⋙ adj.toEquivalence.symm.functor).obj Y ⟶ Z) :
      Y ⟶ ((adj.toEquivalence.symm.inverse ⋙ ihom (F.obj X)) ⋙ G).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}
                  (adj.toEquivalence.symm.functor.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) ⟶ adj.toEquivalence.symm.inverse.obj Z)) :
              (a : adj.toEquivalence.symm.functor.obj (F.obj X ⊗ F.obj Y) ⟶ Z) →
                F.obj X ⊗ F.obj Y ⟶ adj.toEquivalence.symm.inverse.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)))
⊢ Eq (α := Y ⟶ ((adj.toEquivalence.symm.inverse ⋙ ihom (F.obj X)) ⋙ G).obj Z)
    ((Adjunction.homEquiv
            (Adjunction.ofNatIsoLeft
              (adj.comp (Adjunction.comp (ihom.adjunction (F.obj X)) adj.toEquivalence.symm.toAdjunction))
              (Iso.compInverseIso (Functor.Monoidal.commTensorLeft F X)))
            Y Z :
          (a : (tensorLeft X).obj Y ⟶ Z) → Y ⟶ ((adj.toEquivalence.symm.inverse ⋙ ihom (F.obj X)) ⋙ G).obj Z)
        f :
      Y ⟶ ((adj.toEquivalence.symm.inverse ⋙ ihom (F.obj X)) ⋙ G).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}
                  (adj.toEquivalence.symm.functor.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) ⟶ adj.toEquivalence.symm.inverse.obj Z)) :
              (a : adj.toEquivalence.symm.functor.obj (F.obj X ⊗ F.obj Y) ⟶ Z) →
                F.obj X ⊗ F.obj Y ⟶ adj.toEquivalence.symm.inverse.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.homEquiv_ofNatIsoLeft_apply]

Premises used: CategoryTheory.Adjunction.homEquiv_ofNatIsoLeft_apply