# Coproducts

We said that a product of $A$ and $B$ was a terminal object in the subcategory induced by the collection of objects that have morphisms to both $A$ and $B$. Then the dual notion would be a called a coproduct, and would be an initial object in the subcategory induced by the collection of objects that have morphisms from both $A$ and $B$.

In other words, the coproduct $A\coprod B$ has injection maps $i_A:A\to A\coprod B$ and $i_B:B\to A\coprod B$, and if there is any other $C$ and $f_A:A\to C$ and $f_B:B\to C$, then there is a unique morphism $g$ which makes

commute.

Let me juxtapose the diagrams for products and coproducts so you can really see the duality.

(a) is a diagram commonly used to describe products. (b) is a diagram commonly used to explain coproducts.

Notice that even the symbol for coproducts is the symbol $\prod$ (commonly used for products) upside down. For whatever reason, some people still use $\times$ for products sometimes. Now that I think about it, I should go back and change all of the $\times$ to $\prod$. Oh well. Laziness prevails.

I should also note that, since we showed initial and terminal objects are unique up to isomorphism, so are coproducts and products.