Homomorphism set to direct product is Cartesian product of homomorphism sets

For groups
Suppose $$A,B,C$$ are groups. Denote by $$B \times C$$ the external direct product of the groups $$B$$ and $$C$$. Denote by $$\operatorname{Hom}(A,B)$$, $$\operatorname{Hom}(A,C)$$, and $$\operatorname{Hom}(A,B \times C)$$ the homomorphism sets between the pairs of groups. There is a canonical bijection:

$$\operatorname{Hom}(A,B) \times \operatorname{Hom}(A,C) \leftrightarrow \operatorname{Hom}(A,B \times C)$$

The bijection is defined as:

$$(\alpha,\beta) \mapsto (g \mapsto (\alpha(g),\beta(g))$$