Is an arrow $f:X\to A\times B$ in a category with products always a pair of arrows $(g,h)$ with $g:X\to A$ and $h:Z\to B$? So if we have such an arrow $f:X\to A\times B$, can we always say that $f=(g,h)$ for some $g,h$?
I know that if you have two arrows $g:X\to A$ and $h:Z\to B$ then the universal property of the product gives an arrow $f:X\to A\times B$, but does the converse hold? Thanks.
For any map $f:X\to A\times B$ we have that $f=(\pi_0 f, \pi_1 f)$. By the definition of the $(-,-)$ notation, we have that $\pi_0(\pi_0 f,\pi_1 f)=\pi_0 f$, and similarly $\pi_1(\pi_0 f,\pi_1 f)=\pi_1 f$. By the universal property of the products, $(\pi_0 f,\pi_1 f)$ is the unique map that yields these morphisms when composed with the projections; but obviously $f$ does too. So we must have $f=(\pi_0 f,\pi_1 f)$.