In the book Conceptual mathematics, by Lawvere et al. it says on page $45$:
When $B$ is a one-element set, then the possibility of factoring a given $h : A \to C$ across $B$ is a very drastic restriction on $h$.
What does it mean "to factor something across"? I believe that this terminology wasn't explained before. And why is it called like that?
"Factor across" here just means solving the "determination" or "choice" problem referred to in the text. In this case, $h$ "factors across" $B$ if there exists $g:B\to C$ such that $h=g\circ f$, where $f:A\to B$ is the unique map. The term "factor" here comes from thinking of composition as "multiplication": you are asking whether you can "factor" $h$ as a "product" $g\circ f$.
(I must admit that I have not encountered the exact phrasing "factor across" before. In my experience, the phrasing "factor through" is much more common.)