I hope everyone is well. A few days ago I was studying a bit about homological algebra and in an article they showed the following:
''Given the finitely generated $R-$modules $L_i,$ $i \in I,$ and the arbitrary $R-$modules $W_j,$ $j \in J,$ we have the following isomorphisms :
$$\prod_{i \in I} {\rm Hom} _R \left(L_i , \bigoplus_{j \in J} X_j \right) \cong\prod_{i \in I} \bigoplus_{ j \in J } {\rm hom} _R \left(L_i , X_j \right)$$
The article doesn't show details of the proof, however, it caused me doubts about the veracity of the statement, does anyone know of a book where it shows a sketch of the proof of the statement?
Something that I know is true is: $${\rm Hom} _R \left( \bigoplus _{i \in I} L_i , \bigoplus _{j \in J} X_j \right) \cong \prod _{i \in I} {\rm Hom} _R \left(L_i , \bigoplus _{j \in J} X_j \right)$$
but i don't know if it is useful in the proof.