I am reading the notes of Pierre Shapira on Algebra and Geometry. He writes the following: Let $A$ be a $k$-algebra, $M \in Mod(A)$. Then $\mathrm{Hom}_A(M, \prod_i N_i) \cong \prod_i \mathrm{Hom}_A(M, N_i)$ and $\mathrm{Hom}_A (\bigoplus_i M_i, N) \cong \prod_i \mathrm{Hom}_A(M_i, N)$. I understand why these are true.
Nevertheless, I was wondering whether the other relations are true as well: $$\textstyle \mathrm{Hom}_A(\prod_i M_i, N) \cong \prod_i \mathrm{Hom}_A(M_i, N)$$ and $$\textstyle \mathrm{Hom}_A (M, \bigoplus_i N_i) \cong \prod_i \mathrm{Hom}_A(M, N_i)$$ If the set of indices I is finite, of course it is because then the natural injection $\bigoplus_i M_i \to \prod_i M_i$ is a bijection (as the author writes on page 12). If not, I think that it is true as well, by the following proof:
For $\mathrm{Hom}_A(\prod_i M_i, N) \cong \prod_i \mathrm{Hom}_A(M_i, N)$: Define $\Phi: \mathrm{Hom}_A(\prod_i M_i, N) \to \prod_i \mathrm{Hom}_A(M_i, N)$, $(\phi: (m_1, \cdots, m_k) \mapsto n = \sum_i \alpha_i m_i) \to (\phi_1: m_1 \mapsto \alpha_i m_i, \cdots, \phi_N: m_k \mapsto \alpha_k m_k)$
This map is surjective, injective and k-linear.
The other one has a similar proof. Is the first statement and proof correct?
No to both questions.
There is a natural homomorphism $$\textstyle \alpha : \mathrm{Hom}_A(\prod_i M_i, N) \to \prod_i \mathrm{Hom}_A(M_i, N), \quad f \mapsto (f \circ \iota_i)_i,$$ but this has no reason to be injective or surjective. In fact, the right hand side is $\mathrm{Hom}_A(\bigoplus_i M_i,N)$, so $\alpha$ corresponds to the map $$\textstyle \iota^*: \mathrm{Hom}_A(\prod_i M_i, N) \to \mathrm{Hom}_A(\bigoplus_i M_i,N)$$ induced by the inclusion $\iota : \bigoplus_i M_i \hookrightarrow \prod_i M_i$. So the Yoneda Lemma tells us that $\alpha$ is an isomorphism for all $N$ if and only if $\iota$ is an isomorphism, which is wrong when $M_i \neq 0$ for infinitely many indices $i$.
For a specific counterexample (you find this by applying the Yoneda proof), take $N = \bigoplus_i M_i$, then $(\iota_i)_i$ will (usually) not be in the image of $\alpha$.
Similarly, there is a natural homomorphism $$\textstyle \beta :\mathrm{Hom}_A (M, \bigoplus_i N_i) \to\prod_i \mathrm{Hom}_A(M, N_i),\quad f \mapsto (p_i \circ f)_i,$$ which is injective, but it has no reason to be surjective, since it corresponds to $$\textstyle \iota_* : \mathrm{Hom}_A (M, \bigoplus_i N_i) \to \mathrm{Hom}_A(M, \prod_i N_i)$$ for the inclusion $\iota$ above, and again the Yoneda Lemma tells us that this is an isomorphism of all $N$ if and only if $\iota$ is an isomorphism.
For a specific counterexample, take $M = \prod_i N_i$, then $(p_i)_i$ will (usually) not be in the image of $\beta$.