Let $\{M_i\}_{i=1}^n$ y $\{N_j\}_{j=1}^m$ two families of $R$-modules. Show that $$Hom_R \left(\bigoplus_{i=1}^n M_i,\bigoplus_{j=1}^m N_j \right) \cong \bigoplus_{i,j}Hom_R(M_i,N_j)$$ Dem: Let $\varphi \in \bigoplus_{i,j}Hom_R(M_i,N_i)$, then $\varphi=(\varphi_{i,j})$, where $\varphi_{i,j}:M_i \rightarrow N_j$, $1\leq i \leq n$ y $1 \leq j \leq m $ we can consider the matrix with entries $\varphi_{i,j}$, and an element $x:=(x_1,\cdots,x_n)$, with $x_i \in M_i$, tal que:
\begin{matrix} \varphi(x)= & \begin{pmatrix} \varphi_{11} &\cdots & \varphi_{n1}\\ \vdots & \ddots & \vdots\\ \varphi_{1m} & \cdots & \varphi_{nm} \end{pmatrix} &\begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix} &= \begin{pmatrix} \sum_{i=1}^n \varphi_{i1}(x_i)\\ \vdots\\ \sum_{i=1}^n \varphi_{im}(x_i) \end{pmatrix} \end{matrix}
Then we define a $R$-homomorphism of modules $\psi : \oplus _{i,j} Hom_R(M_i,N_j) \rightarrow Hom_R(\oplus_i M_i, \oplus_j N_j)$
\begin{matrix}\psi((\varphi_{i,j})+(\tau_{i,j}))(x_1,\ldots,x_n)=\begin{pmatrix} \sum_{i=1}^n (\varphi_{i1}+\tau_{i1})(x_i)\\ \vdots\\ \sum_{i=1}^n (\varphi_{im}+\tau_{im})(x_i) \end{pmatrix}=\begin{pmatrix} \sum_{i=1}^n \varphi_{i1}(x_i)\\ \vdots\\ \sum_{i=1}^n \varphi_{im}(x_i) \end{pmatrix} + \begin{pmatrix} \sum_{i=1}^n \tau_{i1}(x_i)\\ \vdots\\ \sum_{i=1}^n \tau_{im}(x_i) \end{pmatrix} \end{matrix}
I think this is the "natural" homomorphism, but I'm struggling to prove that is a biyection. Or to see the inverse to compose the identity, any hint?
I think you already proved it. Just in case saying here in my answer,
Note that, $n \in \oplus_i N_i$ can be uniquely written as $n = \sum_i n_i$, $n_i \in N_i$.
Take $\phi \in Hom(\oplus_i M_i, \oplus_i N_i)$, then for $m_j \in M_j$,
$\phi(m_j) \in \oplus_i N_i$ hence can be uniquely defined as $\phi(m_j) = \sum_i n_i$.
This uniquely gives a map $\phi_{ji}$ for every fixed $j$ defined as $\phi_{ji}(m_j) = n_i$. Since the map defined is unique, this is an injection between $Hom(\oplus_i M_i,\oplus_i N_i)$ into $\oplus_{ij} Hom(M_i,N_j)$.
Surjection can be proved by taking any maps $\psi_{ij}$ from $\oplus_{ij} Hom(M_i,N_j)$ and extending it to $Hom(\oplus_i M_i,\oplus_i N_i)$ by defining $\psi(m_j) = \sum_{ji} \psi_{ji}(m_j) $.