Given the (external) direct sum of $R$-modules: $V= \bigoplus V_i, W= \bigoplus W_j$, will $\hom(V,W) \cong \bigoplus_{i,j} \hom(V_i,W_j)?$
I know that $\hom(V,\bigoplus_j W_j) \cong \bigoplus_j \hom(V,W_j)$, the isomomorphism given by $\Phi(f) \mapsto (f\circ \pi_1,\cdots, f\circ \pi_s)$. Would then the same argument give us $\hom(V,W_j) \cong \bigoplus_i \hom(V_i,W_j) $? I'm not sure about this, since a $f\in \hom(V,W_j)$ is given by $f(v_1,\cdots, v_k) = w_j \in W_j.$
How to proceed with this? Is it actually true?
In the case where both sums are finite, then you are exactly right (As Chris points out). However, as Lord Shark points out, if either of the sums are infinite things can get weird. In general:
$$\text{Hom} \left ( \bigoplus_i V_i, \prod_j W_j \right ) \cong \prod_i \prod_j \text{Hom}(V_i, W_j)$$
However, since direct sums and direct products coincide when the index set is finite, your answer is correct in that case.
As for why...
Say $f : \bigoplus_i V_i \to \prod_j W_j$.
Then $f \circ \iota_i : V_i \to \prod_j W_j$ (where $\iota_i : V_i \to \bigoplus_i V_i$)
Then $\pi_j \circ f \circ \iota_i : V_i \to W_j$ (where $\pi_j : \prod_j W_j \to W_j$).
Thus, pairing these up, $(\pi_j \circ f \circ \iota_i)_{i, j} : \prod_i \prod_j \text{Hom}(V_i, W_j)$.
Since this construction is uniform in $f$, it defines one direction of the desired isomorphism. I will leave it to you to prove the other direction.
I hope this helps ^_^