The follwing are extracts taken from Tammo tom Dieck's Algebraic Topology, 2008 European Mathematical Society:
My question is the following: I suppose that the category of G-sets contains the empty set, but for $F=\varnothing$, we have $E\times_G F=\varnothing$, so the associated "covering" map $p_F: E\times_G F \to B$ is not surjective (unless $B=\varnothing$); thus, it is not a covering map.
When defining locally trivial maps and coverings, tom Dieck says
We assume that $p$ is surjective to avoid empty fibres.
Do we need to drop this assumption to obtain the desired functor?
Thank you in advance!