bijective correspondence involing homomorphsims to a direct product of groups? Let $G$, $G'$ and $H$ be groups. Establish a bijective correspondence between homomorphisms $\Phi: H \to G \times G'$ from $H$ to the product group and pairs $(\varphi, \varphi')$ consisting of a homomorphism $\varphi: H \to G$ and a homomorphism $\varphi': H \to G'$.
I am confused as to what a bijective correspondence is referring to in this context.

I think what you mean is this : Consider two homomorphisms $\varphi : H\to G$ and $\varphi' : H\to G'$, you get a map
$$
\Phi : H\to G\times G' \text{ given by } h \mapsto (\varphi(h), \varphi'(h))
$$
Conversely, given a map $\Phi : H\to G\times G'$, consider the projection homomorphisms
$$
\pi_1 : G\times G' \to G \text{ and } \pi_2 : G\times G'\to G'
$$
and set
$$
\varphi = \pi_1\circ \Phi, \text{ and } \varphi' = \pi_2\circ \Phi
$$
Then $\varphi$ and $\varphi'$ are group homomorphisms, and
$$
\Phi(h) = (\varphi(h), \varphi'(h)) \quad\forall h\in H
$$
This gives a bijection
$$
\text{Hom}(H,G\times G') \to \text{Hom}(H,G)\times \text{Hom}(H,G')
$$
