In the category Ab, the product and coproduct coincide $(C_1 \times C_2 = C_1 \oplus C_2 = C_1 \coprod C_2)$?

In Ab, the product and coproduct coincide $(C_1 \times C_2 = C_1 \oplus C_2 = C_1 \coprod C_2)$

I'm not exactly sure how I'd show these two objects in the category Ab are the same.

I've tried using the commutative diagrams from the definition of product and coproduct, but I can't see how to show this equality.

Anyone have any ideas?

• Obviously you'll need to use some properties of the category itself, not just the definitions of product/coproduct, as it is not true in a general category. – Dan Rust Sep 22 '17 at 12:36

Let $Z$ be a product of two objects $X,Y$ in a pre-additive category with projections $p:Z\to X$ and $q:Z\to Y$. By universal property of product, there exists $i:X\to Z$ and $j:Y\to Z$ such that $ip=1_X$, $iq=0$ and $jp=0$ an $jq=1_Y$. Let $h=pi+qj:Z\to Z$. Then $hp=p$ and $hq=q$ thus $h=1_Z$.
We claim that $Z$ is a coproduct of $X,Y$ with inclusions $i,j$. For let $x:X\to W$ and $y:Y\to W$. Then $z=px+qy$ is the only morphism such that $iz=x$ and $jz=y$.