-- Mathlib/CategoryTheory/Shift/Induced.lean
@[simp]
lemma add_inv_app_obj (a b : A) (X : C) :
    (add F s i a b).inv.app (F.obj X) =
      (s b).map ((i a).hom.app X) ≫ (i b).hom.app ((shiftFunctor C a).obj X) ≫
        F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X

Proof state:
C : Type u_4
D : Type u_2
inst✝⁵ : Category.{u_3, u_4} C
inst✝⁴ : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
a b : A
X : C
⊢ (add F s i a b).inv.app (F.obj X) =
    (s b).map ((i a).hom.app X) ≫
      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X

Tactic:
have h : whiskerLeft F (add F s i a b).inv = _ :=
  ((whiskeringLeft C D D).obj F).map_preimage _

Result:
C : Type u_4
D : Type u_2
inst✝⁵ : Category.{u_3, u_4} C
inst✝⁴ : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
a b : A
X : C
h :
  whiskerLeft F (add F s i a b).inv =
    (i (a + b) ≪≫
        isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫
          (shiftFunctor C a).associator (shiftFunctor C b) F ≪≫
            isoWhiskerLeft (shiftFunctor C a) (i b).symm ≪≫
              ((shiftFunctor C a).associator F (s b)).symm ≪≫
                isoWhiskerRight (i a).symm (s b) ≪≫ F.associator (s a) (s b)).inv
⊢ (add F s i a b).inv.app (F.obj X) =
    (s b).map ((i a).hom.app X) ≫
      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X

Proof state:
C : Type u_4
D : Type u_2
inst✝⁵ : Category.{u_3, u_4} C
inst✝⁴ : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
a b : A
X : C
h :
  whiskerLeft F (add F s i a b).inv =
    (i (a + b) ≪≫
        isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫
          (shiftFunctor C a).associator (shiftFunctor C b) F ≪≫
            isoWhiskerLeft (shiftFunctor C a) (i b).symm ≪≫
              ((shiftFunctor C a).associator F (s b)).symm ≪≫
                isoWhiskerRight (i a).symm (s b) ≪≫ F.associator (s a) (s b)).inv
⊢ (i (a + b) ≪≫
            isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫
              (shiftFunctor C a).associator (shiftFunctor C b) F ≪≫
                isoWhiskerLeft (shiftFunctor C a) (i b).symm ≪≫
                  ((shiftFunctor C a).associator F (s b)).symm ≪≫
                    isoWhiskerRight (i a).symm (s b) ≪≫ F.associator (s a) (s b)).inv.app
      X =
    (s b).map ((i a).hom.app X) ≫
      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X

Tactic:
simp

Result:
no goals

Proof state:
C : Type u_4
D : Type u_2
inst✝⁵ : Category.{u_3, u_4} C
inst✝⁴ : Category.{u_1, u_2} D
F : C ⥤ D
A : Type u_5
inst✝³ : AddMonoid A
inst✝² : HasShift C A
s : A → D ⥤ D
i : (a : A) → F ⋙ s a ≅ shiftFunctor C a ⋙ F
inst✝¹ : ((whiskeringLeft C D D).obj F).Full
inst✝ : ((whiskeringLeft C D D).obj F).Faithful
a b : A
X : C
h :
  whiskerLeft F (add F s i a b).inv =
    (i (a + b) ≪≫
        isoWhiskerRight (shiftFunctorAdd C a b) F ≪≫
          (shiftFunctor C a).associator (shiftFunctor C b) F ≪≫
            isoWhiskerLeft (shiftFunctor C a) (i b).symm ≪≫
              ((shiftFunctor C a).associator F (s b)).symm ≪≫
                isoWhiskerRight (i a).symm (s b) ≪≫ F.associator (s a) (s b)).inv
⊢ (add F s i a b).inv.app (F.obj X) =
    (s b).map ((i a).hom.app X) ≫
      (i b).hom.app ((shiftFunctor C a).obj X) ≫ F.map ((shiftFunctorAdd C a b).inv.app X) ≫ (i (a + b)).inv.app X

Tactic:
exact (NatTrans.congr_app h X).trans (by simp)

Result:
no goals