We work in an abelian category. I am looking for a way to prove the isomorphism
$$ \text{Ext}^1\left(\bigoplus_{k=1}^m C_k, \bigoplus_{j=1}^n A_j \right) \cong \bigoplus_{k=1}^m \bigoplus_{j=1}^n \text{Ext}^1\left(C_k, A_j \right)$$
using the interpretation of $\text{Ext}^1\left(C_k, A_j \right)$ as equivalence classes of extensions
$$ 0 \to A \to B \to C \to 0.$$
In particular, I would like to know explicitly how the bijection is defined (in both directions). I think that one sends an extension on the LHS to the tuple of extensions obtained doing pullback by the inclusions $C_i \to \oplus_k C_k$ and pushout by the projections $\oplus_j A_j \to A_\ell$. But I cannot see why this is an isomorphism, nor what would be the inverse map.