Counterexamples in homological algebra

Question

I search for a counterexample for modules such that $\mathrm{Hom} (\prod M_i,N)$ is not isomorphic to $\oplus \mathrm {Hom}(M_i,N)$ as abelian groups. Also, a counterexample such that $\mathrm {Hom} (M, \oplus N_i)$ be not isomorphic to $\prod \mathrm {Hom} (M, N_i)$ as abelian groups. I do aware of the well-known isomorphisms in classical algebra texts, such as the one replacing direct sum and direct product in the first question, or changing the direct sum to direct product in the second. Thanks for any reply!