Let $Z$, $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous surjection then is the induced map $g \to f\circ g$ from $C(Z,X)$ to $C(Z,Y)$ is continuous. But is it still a surjection?


My issue is that it's not clear if it has a right-inverse...



Surjection in compact-open topology
In many cases, $f\_\ast: C(Z,X)\to C(Z,Y)$, $g\mapsto f\circ g$ is not surjective: Put $Z=Y$ and $h=id\_Y \in C(Z,Y)$, then $h$ is in the range of $f\_\ast$ only if $f$ has a right inverse.


