Let $f: X \to Y$ and $g: Y \to Z$ be quotient maps. The idea is to show that the composite function $g \circ f: X \to Z$ is a quotient map. To do this, we must proof that $V^{\text{open}} \subseteq Z$ if and only if $(g \circ f)^{-1}(V)$ is open in $X$.
For the first implication:
Let $f: X \to Y$ and $g: Y \to Z$ be quotient maps, in particular, both continuous functions. If $V^{\text{open}} \subseteq Z$, then $(g \circ f)^{-1}(V)$ is open in $X$ due to the continuity of $f$ and $g$.
conversely:
Similarly, let $f: X \to Y$ and $g: Y \to Z$ be quotient maps. Furthermore, if $(g \circ f)^{-1}(V)$ is open in $X$, this implies that $$(g \circ f)^{-1}(V)=f^{-1}(g^{-1}(V))$$ is open.
Since $f$ is a quotient function, it follows that $$g^{-1}(V)$$ is open.
Moreover, as $g$ is a quotient function, it must be the case that $V$ is open in $Z$
Is there anything to improve this? any suggestions? thanks!