Is there a slick proof that precompactness is preserved under continuous images?
By slick I mean analogous to the following proof that compactness is preserved under continuous images. Is it even true in complete generality?
Take an open cover of the image of $f$. Its pullback covers the domain of $f$, hence has a finite subcover of pullbacks. The corresponding opens pulled back are a finite open cover because of the counit $f_\ast f^\ast \Rightarrow 1$.