The universal properties of products and coproducts "amount" to the statements
$$ \hom(\coprod_i X_i , Y) = \prod_i \hom(X_i, Y) \quad \text{and} \quad \hom(X,\prod_i Y_i) = \prod_i \hom(X,Y_i)$$
for any category and any objects for which the (co)product is defined. I am wondering if there is any general statement about the other two cases: are there any similar formulas that describe $\hom(X,\coprod_i Y_i)$ and $\hom(\prod_i X_i , Y)$? I don't expect a general formula to exist, but maybe one can say something at least for abelian categories?