Section adj_def.
Variables (C D : Category) (F : Functor D C) (G : Functor C D).
SubClass Adj := NatIso (FunSET2_r F) (FunSET2_l G).
Variables (phi : NT (FunSET2_r F) (FunSET2_l G))
(phi_1 : NT (FunSET2_l G) (FunSET2_r F)).
Variable phi_iso : AreNatIsos phi phi_1.
Definition Build_Adj : Adj := Build_NatIso phi_iso.
Variables (d : D) (c : C) (ad : Adj).
Definition ApAphi (f : F d --> c) : d --> G c := ad (Build_POb1 d c) f.
Definition ApAphi_inv (g : d --> G c) : F d --> c :=
NatIso_inv ad (Build_POb1 d c) g.
(* rewrite rules *)

Lemma Aphi_invPres :
forall f g : d --> G c, f =_S g -> ApAphi_inv f =_S ApAphi_inv g.