Hom turns sums into products

Hom out of a direct sum canonically identifies with the product of Homs.

Hom turns sums into products

Hom turns sums into products: Let $\{M_i\}_{i\in I}$ be a family of $R$ -modules and let $N$ be an $R$ -module. Restriction along the canonical maps $\iota_i:M_i\to \bigoplus_{i\in I}M_i$ induces a natural isomorphism

\operatorname{Hom}_R\Bigl(\bigoplus_{i\in I} M_i,\,N\Bigr)\;\cong\;\prod_{i\in I}\operatorname{Hom}_R(M_i,N).

This expresses the Hom functor as converting a direct sum in the source into a direct product of hom-sets, complementing how tensor product behaves with sums.