Direct sum and Hom Let $R$ be a commutative ring with unity and let $M$ be a finitely generated noetherian R-module. 
Can someone tell me how the isomorphism 
$$
Hom_R(R\oplus R, M)\simeq Hom_R(R,M)\oplus Hom_R(R,M)\simeq M\oplus M
$$
is given? I know how the second isomorphism is given but I don't know about the first. Where does it send a map $R\oplus R\to M$ to?

The universal property of the biproduct is precisely the existence of the first isomorphism (assuming it is natural in $M$).
More generally, the coproduct of $X \amalg Y$ of two objects in any category is the object that satisfies an isomorphism natural in $Z$:
$$ \hom(X \amalg Y, Z) \cong  \hom(X, Z) \times \hom(Y, Z) $$
In the case of $R$-modules, finite products and coproducts are both given by the biproduct; i.e. $X \amalg Y \cong X \oplus Y$ and $M \times N \cong M \oplus N$.
The projection
$$ \hom(X \amalg Y, Z) \to \hom(X, Z) $$
is precisely the map induced by the insertion $i_1 : X \to X \amalg Y$. That is, it sends a function $g : X \amalg Y \to Z$ to the function $g \circ i_1 : X \to Z$.
The same is true for the other projection. The isomorphism from the natural property is the map $g \mapsto (g \circ i_1, g \circ i_2)$.
