Proof:

It is easy to see that $\phi$ is well-defined, bijective and satisfies $\phi\circ \eta = f$. Since $f$ is continuous and $\eta$ is a quotient map we see by the universal property that $\phi$ is continuous. Since $f$ is a quotient map and $\eta$ is continuous we may invoke the universal property again but this time to $\phi^{-1}\circ f = \eta$ to conclude that $\phi^{-1}$ is continuous as well.