End FullImage.
+(* any functor may be restricted to a subcategory of its domain *)
+Section RestrictDomain.
+
+ Context `{C:Category}.
+ Context `{D:Category}.
+ Context `(F:!Functor C D fobj).
+ Context {Pmor}(S:WideSubcategory C Pmor).
+
+ Instance RestrictDomain : Functor S D fobj :=
+ { fmor := fun a b f => F \ (projT1 f) }.
+ intros; destruct f; destruct f'; simpl in *.
+ apply fmor_respects; auto.
+ intros. simpl. apply fmor_preserves_id.
+ intros; simpl; destruct f; destruct g; simpl in *.
+ apply fmor_preserves_comp.
+ Defined.
+
+End RestrictDomain.
+
(*
Instance func_opSubcat `(c1:Category)`(c2:Category)`(SP:@SubCategory _ _ c2 Pobj Pmor)
{fobj}(F:Functor c1⁽ºᑭ⁾ SP fobj) : Functor c1 SP⁽ºᑭ⁾ fobj :=