Map dropping (or lifting?) to another

Question

If I have a map $f:X\to Y$, a quotient map $\pi:X\to Z$, and a map $g:Z\to Y$ such that $f = g \circ \pi$, which of the following statements are appropriate?

$\bullet$ $f$ lifts to $g$

$\bullet$ $f$ drops to $g$

$\bullet$ $g$ lifts to $f$

$\bullet$ $g$ drops to $f$

Much appreciated!

Answer 1

I've never heard "drops" used as a technical term.

Lifts refers to "lifting over the target/codomain": If you have a quotient mapping $h:\tilde{Y} \to Y$ (such as a covering map) and $\tilde{f}:X \to \tilde{Y}$ satisfies $f = h \circ \tilde{f}$, then $\tilde{f}$ is a lift of $f$.

In your situation, one says $f$ factors through $\pi$ (and has induced map $g$), or perhaps that $g$ is induced by $f$ mod $\pi$.