Homomorphism set to direct product is Cartesian product of homomorphism sets

Statement

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:

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

For algebras in an arbitrary variety

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