Definition: Lift

In mathematics, given the morphisms $f: X$→$Y$ and $g: Y$→$Z$, a lift of $f$ to $Z$ is a morphism $h: X$→$Z$ such that $f$ = $g(h)$.

The basic idea involves two objects, $X$ and $Y$, that interact intuitively (as required) and make​ it easy to define a good morphism ($f: X$→$Y$). Then, there exists another object, $Z$, that easily relates to $Y$ (morphism $g: Y$→$Z$), but it’s not explicit in how it relates to $X$.

In this case, the morphism $g$ will be used to lift $f$ to $Z$ using $g$. A new morphism ($h: X$→$Z$) will be determined where $f$ = $g(h)$.