Theorem 4.1 (Universal property of quotients):

Suppose that $X$ is a topological space, $Y$ is a set, $f : X \longrightarrow Y$ is a surjective map and $Y$ is assigned the quotient topology induced by $f$. Then given any topological space $Z$ and map $g : Y \longrightarrow Z$, the map $g$ is continuous if and only if $g\circ f : X \longrightarrow Z$ is continuous.

$$\xymatrix{ X \ar[rr]^{f}\ar[rd]_{g\circ f} & & Y \ar[ld]^{g}\\ & Z }$$