# What does it mean to say a map “factors through” a set?

Consider the following diagram:

What does it mean precisely to say “$f$ factors through $G/\text{ker}(f)$“?

Does it mean $f = \tilde{f} \circ \pi$, for some $\tilde{f}$?

I’ve seen texts use the phrase, but never a definition of this notion.

It means exactly what you write: that you can express $f$ as “product” (composition) of two functions, with the first function going through $G/\mathrm{ker}(f)$; by implication, that map will be the “natural” map into the quotient, i.e., $\pi$. Under more general circumstances, you would also indicate the map in question.
The reason for the term “factors” is that if you write composition of functions by juxtaposition, which is fairly common, then the equation looks exactly as if you “factored” $f$: $f=\tilde{f}\pi$.