Problem 13.3 of Probability and Measure by Billingsley states:
$(\Omega, \mathcal{F})$ and (\Omega', \mathcal{F}') are two measurable spaces. Suppose that $f: \Omega \rightarrow \mathbb{R}^1$ and T:\Omega \rightarrow \Omega' . Show that $f$ is measurable T^{-1}\mathcal{F}':= \{ T^{-1}A': A' \in \mathcal{F}' \} if and only if there exists a map \phi: \Omega' \rightarrow \mathbb{R}^1 such that $\phi$ is measurable \mathcal{F}' and $f= \phi T$.
It seems to me that $T$ behaves like some special concept in category theory, but don't know which. so I wonder if the statement indeed can be described using category language? Thanks!