Suppose that $G_1, G_2, G_3, G_4$ be groups and consider $Hom(A, B)$ as collection of group homomorphisms from the group $A$ to the group $B$. We know that in general it is not a group but a set.
I know that there is a bijection from the set $Hom(G_1\times G_2, G_3\times G_4 )$ to the set $Hom(G_1, G_3)\times Hom(G_1, G_4)\times Hom(G_2, G_3)\times Hom(G_2, G_4)$. I have come to know from my seniors that its easy to prove this result using module theory or some thing like which I have zero knowledge.
But I want to establish this using simple elementary set theoratic approach. I mean is it possible to construct any explicit bijection here using elementary group theory only ? I am confused if my question is correct or not and if correct, then how to complete it ?
Thanks in advance