In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map $f : Y\rightarrow X$, then does the cover $C\times_X Y\rightarrow Y$ correspond to the (equivalence class of) representation $\pi_1(Y)\stackrel{f_*}{\longrightarrow}\pi_1(X)\rightarrow S_n$? (with suitably chosen base points, and assuming $X,Y$ are path connected).
I feel like this should be the case, and it is the case in every example I've worked out, though I can't seem to find this fact anywhere.
If this is false, are there suitable, hopefully tame requirements on $X,Y,f$ for this to be true?