Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base.


What are the relations between the following pairs?


$$f\text{ epi}\qquad f^\ast \text{ faithful}$$


$$f\text{ mono}\qquad f^\ast \text{ full}$$


$$f\text{ strong epi}\qquad f^\ast \text{ conservative}$$


I don't mind completeness assumptions. Just trying to work out whether the behavior of the codomain fibration is "general".


Is there perhaps a reference for results of this kind?



In a fibration, how do properties of arrows downstairs affect the base-change functors?
In short: There is absolutely no such relations, and in general essentially no properties of arrows of the codomain (except being a split epi/split mono or an iso) have any effect on the base change functor.


Indeed, because of Grothendieck's construction, absolutely any (pseudo)-functor from $X^{op}$ to $Cat$ is obtained as the base change functor corresponding to a Fibration of codomain $X$. 


So you can send you epis and mono on whatever you like, you are asking if a completely general functor send epi or mono to some specific classes of maps. The only thing that are preserved are those that are expressed using justs "equations", so: "isomorphisms", "split epis" and "split monos" and that is essentially it.


