-- Module: Mathlib.CategoryTheory.Shift.Adjunction
Goal:
C : Type u_1
D : Type u_2
inst : Category.{v_1, u_1} C
inst_1 : Category.{v_2, u_2} D
F : Functor.{v_1, v_2, u_1, u_2} C D
G : Functor.{v_2, v_1, u_2, u_1} D C
adj : F ⊣ G
A : Type u_3
inst_2 : AddGroup A
inst_3 : HasShift.{v_1, u_1, u_3} C A
inst_4 : HasShift.{v_2, u_2, u_3} D A
a : A
inst_5 : F.CommShift A
Y : D
h : -a + a = 0
simp :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y) ⟶
      G.obj (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) Y))
    (adj.unit.app (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)) ≫
      G.map
          (NatTrans.app (Iso.inv (shiftFunctorCompIsoId D (-a) a ⋯))
            (F.obj (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)))) ≫
        CategoryStruct.comp (Y :=
          G.obj
            (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a)
              (F.obj
                (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C (-a))
                  (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y))))))
          (G.map
            (Functor.map (shiftFunctor D a)
              (NatTrans.app (Iso.inv (Functor.commShiftIso F (-a)))
                (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)))))
          (CategoryStruct.comp (Y :=
            G.obj
              (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a)
                (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D (-a))
                  (F.obj (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y))))))
            (G.map
              (Functor.map (shiftFunctor D a)
                (NatTrans.app (Iso.hom (Functor.commShiftIso F (-a)))
                  (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)))))
            (G.map
                (Functor.map (shiftFunctor D a)
                  (Functor.map (shiftFunctor D (-a)) (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) (G.obj Y)))) ≫
              G.map
                  (Functor.map (shiftFunctor D a)
                    (NatTrans.app (Iso.hom (shiftFunctorCompIsoId D a (-a) ⋯)) (F.obj (G.obj Y)))) ≫
                G.map (Functor.map (shiftFunctor D a) (adj.counit.app Y)))))
    (adj.unit.app (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)) ≫
      G.map
          (NatTrans.app (Iso.inv (shiftFunctorCompIsoId D (-a) a h))
            (F.obj (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)))) ≫
        G.map
            (Functor.map (shiftFunctor D a)
              (Functor.map (shiftFunctor D (-a)) (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) (G.obj Y)))) ≫
          G.map
              (Functor.map (shiftFunctor D a)
                (NatTrans.app (Iso.hom (shiftFunctorCompIsoId D a (-a) ⋯)) (F.obj (G.obj Y)))) ≫
            G.map (Functor.map (shiftFunctor D a) (adj.counit.app Y)))
⊢ Eq (α := (G ⋙ shiftFunctor C a).obj Y ⟶ (shiftFunctor D a ⋙ G).obj Y) (NatTrans.app (Iso.inv (iso adj a)) Y)
    (adj.unit.app (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)) ≫
      G.map
          (NatTrans.app (Iso.inv (shiftFunctorCompIsoId D (-a) a h))
            (F.obj (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj Y)))) ≫
        G.map
            (Functor.map (shiftFunctor D a)
              (Functor.map (shiftFunctor D (-a)) (NatTrans.app (Iso.hom (Functor.commShiftIso F a)) (G.obj Y)))) ≫
          G.map
              (Functor.map (shiftFunctor D a)
                (NatTrans.app (Iso.hom (shiftFunctorCompIsoId D a (-a) ⋯)) (F.obj (G.obj Y)))) ≫
            G.map (Functor.map (shiftFunctor D a) (adj.counit.app Y)))

Tactic:
simp only [iso, iso', shiftEquiv', Equiv.toFun_as_coe, conjugateIsoEquiv_apply_inv,
    conjugateEquiv_apply_app, Functor.comp_obj, comp_unit_app, Functor.id_obj,
    Equivalence.toAdjunction_unit, Equivalence.Equivalence_mk'_unit, Iso.symm_hom, Functor.comp_map,
    comp_counit_app, Equivalence.toAdjunction_counit, Equivalence.Equivalence_mk'_counit,
    Functor.map_shiftFunctorCompIsoId_hom_app, assoc, Functor.map_comp]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, CategoryTheory.conjugateEquiv_apply_app, Equiv.toFun_as_coe, CategoryTheory.shiftEquiv', CategoryTheory.Iso.symm_hom, CategoryTheory.Adjunction.comp_counit_app, CategoryTheory.Functor.map_shiftFunctorCompIsoId_hom_app, CategoryTheory.Equivalence.toAdjunction_counit, CategoryTheory.Equivalence.Equivalence_mk'_unit, CategoryTheory.Functor.id_obj, CategoryTheory.Adjunction.RightAdjointCommShift.iso', CategoryTheory.conjugateIsoEquiv_apply_inv, CategoryTheory.Functor.map_comp, CategoryTheory.Adjunction.RightAdjointCommShift.iso, CategoryTheory.Equivalence.Equivalence_mk'_counit, CategoryTheory.Functor.comp_map, CategoryTheory.Equivalence.toAdjunction_unit, CategoryTheory.Adjunction.comp_unit_app