What I want to prove is : there does not exist two non-isomorphism group $G_1,G_2$ satisfy $Hom(G_1,F)=Hom(G_2,F)$ for any group $F$, here $Hom$ are hom in the category of all group. Or If you like, the category of all small group $Grp$, i.e. the group whose underlying structure is set. And $Hom(A,B)=Hom(C,B)$ is up to natrual isomorphism (there exist $f_B \in Hom(A,C)$ s.t. for any $p_B \in Hom(A,B)$ there exist $q_B \in Hom(C,B) s.t. q_Bf=p_B$ and conversely there exist $f'_B \in Hom(C,A)$ s.t. $p'_Bf'_B=q'_B$. And with another group $D$, and group homomorphism $k: B\to D$ and $f_D,f'_D$,satisfy $kp_Bf_B=f_Dkp_D$)
When $F={e}$ we know cardinal of $G$. I think Yoneda lemma may help, but as far as I know, the category of all group is not a locally small category. If this is wrong, please give a counter-example
Update: Yoneda lemma indeed solve the $Grp$ case, but I intended to prove for group that is not small.