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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!

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