I need help on a exercise:
Let $M$ be a $A$-module with finite presentation, and $(N_{\lambda})_{\lambda \in \Lambda}$ a family of $A$-modules. Show that exist a functorial isomorphism of $\mathbb{K}$-modules: $$Hom_A \left( M,\bigoplus_{\lambda \in \Lambda}N_{\lambda} \right) \xrightarrow{\sim} \bigoplus_{\lambda \in \Lambda} Hom_A(M,N_{\lambda}).$$
The finite presentation part means that the $L_0$ and $L_1$ from the free presentation of $M$ are of finite type.
This actually holds when $M$ is just assumed to be finitely generated. We have the general isomorphism $\hom(M,\prod_{\lambda} N_{\lambda}) \cong \prod_{\lambda} \hom(M,N_{\lambda})$, $f \mapsto (f_{\lambda})_{\lambda}$. Therefore, it suffices to prove that this isomorphism maps the submodule $\hom(M,\bigoplus_{\lambda} N_{\lambda})$ of $\hom(M,\prod_{\lambda} N_{\lambda})$ onto the submodule $\bigoplus_{\lambda} \hom(M,N_{\lambda})$ of $\prod_{\lambda} \hom(M,N_{\lambda})$. I only show one direction, the other direction is even simpler: Let $E$ be a finite generating set of $M$. If $f : M \to \bigoplus_{\lambda} N_{\lambda}$ is a homomorphism, then for every generator $e \in E$ there is some finite subset of $\Lambda$ such that $f(e)$ lies in the finite direct sum indexed by that subet. Since $E$ is finite, we find a finite subset $\Lambda' \subseteq \Lambda$ such that $f(e)$ lies in $\bigoplus_{\lambda \in \Lambda'} N_{\lambda}$ for all $e \in E$. Since $E$ generates $M$, it follows that $f(M)$ lies in $\bigoplus_{\lambda \in \Lambda'} N_{\lambda}$. But this means that for all $\lambda \in \Lambda \setminus \Lambda'$ the component $f_{\lambda} : M \to N_{\lambda}$ is zero, so that, in fact, $(f_{\lambda})_{\lambda} \in \bigoplus_{\lambda} \hom(M,N_{\lambda})$.