In the category of Abelian Groups, what precisely is the canonical morphism? What about for the category of sets? I would like a relatively concrete answer. Thanks!
Sorry, I need to be more precise. I am trying to show that the category of Sets is distributive. So, given any sets $X, Y,$ and $Z$, I want to know what the canonical morphism is from $(X\times Y)\oplus(X\times Z)\to X\times(Y\oplus Z)$.
 A: Yuri Delanghe's comment is correct, of course, but I am not sure it satisfies your desire for concreteness.  In the case of sets, $(X\times Y)\oplus(X\times Z)$
is the disjoint union of $X\times Y$ and $X\times Z$.
A standard way of making two sets disjoint is multiplying the first set with $\{0\}$ and the second with $\{1\}$.  We use the same technique for $Y\oplus Z$.
Now each element of $(X\times Y)\oplus(X\times Z)$ is of the form $((x,a),i)$
where $a\in Y\cup Z$ and $i\in\{0,1\}$.
The canonical morphism maps this to $(x,(a,i))$, which is in $X\times(Y\oplus Z)$.
In the case of abelian groups, $X\times(Y\oplus Z)$ and $(X\times Y)\oplus(X\times Z)$ are not isomorphic, in general.  There is a nice morphism from
$X\times(Y\oplus Z)$ to $(X\times Y)\oplus(X\times Z)$, though.
Namely, since for finitely many factors product and coproduct coinside with the direct sum of abelian groups, the elements of $X\times(Y\oplus Z)$ are of the form $(x,(y,z))$ with $x\in X$, $y\in Y$, and $z\in Z$.
Such an element gets mapped to $((x,y),(x,z))$ in $(X\times Y)\oplus(X\times Z)$.
