Homomorphism set to direct product is Cartesian product of homomorphism sets

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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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