End MorphismTheory.
Notation "''ker' f" := (ker_group (MorPhantom f)) : Group_scope.
Notation "''ker_' G f" := (G :&: 'ker f)%G : Group_scope.
Notation "f @* G" := (morphim_group (MorPhantom f) G) : Group_scope.
Notation "f @*^-1 M" := (morphpre_group (MorPhantom f) M) : Group_scope.
Notation "f @: D" := (morph_dom_group f D) : Group_scope.
Arguments injmP {aT rT D f}.
Arguments morphpreK {aT rT D f} [R] sRf.
Section IdentityMorphism.
Variable gT : finGroupType.
Implicit Types A B : {set gT}.
Implicit Type G : {group gT}.
Definition idm of {set gT} := fun x : gT => x : FinGroup.sort gT.

Lemma ker_factm_loc (K : {group aT}) : 'ker_(q @* K) ff = q @* 'ker_K f.