Product (part 3)

Here’s another way to think about products. In fact, this is how it’s more commonly described. Let $\mathcal C$ be a category, and $A$ and $B$ objects in $\mathcal C$. The product (if it exists) is some object $A\times B$ with maps $\pi_A$ and $\pi_B$ in the diagram

such that if there is any other object $P$ and maps $f_A:P\to A$ and $f_B:P\to B$, then there is a unique map $g$ which makes the following diagram commute:

The diagram commuting is simply saying that $\pi_A\circ g=f_A$ and $\pi_B\circ g=f_B$. Notice that this exactly the same property that I said last time. I said “it’s the terminal object in the subcategory of things that map into both $A$ and $B$. This is a wonderful exercise to test your understanding of these definitions. I highly suggest working through it.