Definition kAut U V f := kHom U V f && (f @: V == V)%VS.

(* Specifications: *)
Lemma kAutE K E f : kAut K E f = kHom K E f && (f @: E <= E)%VS.
Lemma kAutS U1 U2 V f : (U1 <= U2)%VS -> kAut U2 V f -> kAut U1 V f.
Lemma kAutfE K f : kAut K {:L} f = kHom K {:L} f.
Lemma inv_kHomf K f : kHom K {:L} f -> kHom K {:L} f^-1.
Lemma inv_is_ahom (f : 'AEnd(L)) : ahom_in {:L} f^-1.
Canonical inv_ahom (f : 'AEnd(L)) : 'AEnd(L) := AHom (inv_is_ahom f).