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Let $\{M_{i}\}_{i\in I}$ be a family of $R$-modules and also $N$ is a $R$-module. Is there an counterexample for the following relation: ${\rm Hom}_{R}(\prod_{i\in I}M_{i},N)\cong_{\mathbb{Z}} \coprod_{i\in I}{\rm Hom}_{R}(M_{i},N).$

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    @ZhenLin $\coprod$ mean that the category-theoretic coproduct,2012-11-11

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Take $R=k$ a field, $I$ any infinite set, $M_i = k$ for each $i$, and $N=k$. Then the left hand side is the dual of $k^I$, which has dimension strictly greater than the cardinal of $I$, whereas the right hand side is the direct sum of $I$ copies of the dual of $k$, which has dimension the cardinal of $I$.