-- 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
inst_5 : G.CommShift A
a : A
X : C
simp :
  Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) X ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj (F.obj X)))
    (Functor.map (shiftFunctor C a) (adj.unit.app X))
    (Functor.map (shiftFunctor C a) (adj.unit.app X) ≫
      CategoryStruct.comp (Y :=
        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))
            (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (G.obj (F.obj X)))))
        (Functor.map (shiftFunctor C a) (NatTrans.app (Iso.inv (shiftFunctorCompIsoId C a (-a) ⋯)) (G.obj (F.obj X))))
        (CategoryStruct.comp (Y :=
          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 (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X)))))
          (Functor.map (shiftFunctor C a)
            (Functor.map (shiftFunctor C (-a)) (NatTrans.app (Iso.inv (Functor.commShiftIso G a)) (F.obj X))))
          (CategoryStruct.comp (Y :=
            Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a)
              (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 X)))))
            (Functor.map (shiftFunctor C a)
              (NatTrans.app (Iso.inv (Functor.commShiftIso G (-a)))
                (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X))))
            (Functor.map (shiftFunctor C a)
                (NatTrans.app (Iso.hom (Functor.commShiftIso G (-a)))
                  (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X))) ≫
              NatTrans.app (Iso.hom (shiftFunctorCompIsoId C (-a) a ⋯))
                  (G.obj (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X))) ≫
                NatTrans.app (Iso.hom (Functor.commShiftIso G a)) (F.obj X)))))
⊢ Eq (α :=
    Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) (Functor.obj.{v_1, v_1, u_1, u_1} (𝟭 C) X) ⟶
      Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) ((F ⋙ G).obj X))
    (Functor.map (shiftFunctor C a) (adj.unit.app X))
    (adj.unit.app (Functor.obj.{v_1, v_1, u_1, u_1} (shiftFunctor C a) X) ≫
      CategoryStruct.comp (Y := G.obj ((F ⋙ shiftFunctor D a).obj X))
        (G.map
          (F.map (Functor.map (shiftFunctor C a) (adj.unit.app X)) ≫
            F.map
                (Functor.map (shiftFunctor C a)
                  (G.map (NatTrans.app (Iso.inv (shiftFunctorCompIsoId D a (-a) ⋯)) (F.obj X)))) ≫
              F.map
                  (Functor.map (shiftFunctor C a)
                    (NatTrans.app (Iso.hom (Functor.commShiftIso G (-a)))
                      (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X)))) ≫
                F.map
                    (NatTrans.app (Iso.hom (shiftFunctorCompIsoId C (-a) a ⋯))
                      (G.obj (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X)))) ≫
                  adj.counit.app (Functor.obj.{v_2, v_2, u_2, u_2} (shiftFunctor D a) (F.obj X))))
        (NatTrans.app (Iso.hom (Functor.commShiftIso G a)) (F.obj X)))

Tactic:
simp only [Functor.id_obj, Functor.comp_obj, Functor.map_shiftFunctorCompIsoId_inv_app,
    Functor.map_comp, assoc, unit_naturality_assoc, right_triangle_components_assoc]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, CategoryTheory.Functor.id_obj, CategoryTheory.Functor.map_comp, CategoryTheory.Adjunction.right_triangle_components_assoc, CategoryTheory.Functor.map_shiftFunctorCompIsoId_inv_app, CategoryTheory.Adjunction.unit_naturality_assoc