Generalization of topological functor which need not be fibre-complete Is there a useful property for a concrete category $(\mathcal A, U : \mathcal A \to \mathcal X)$ such that in this case $U$ is fibre-complete iff $U$ is topological? (kind of like what "flat" is too "left exact). 
The motivation is the following: given a finitely complete category $\mathcal X$ we can define the concrete category of $\mathcal X$-objects equipped with relations / reflexive relations / preorders / congruences / etc. . These seem to be topological iff the functor $U : (X,R) \mapsto X$ is fibre-complete (unless I'm mistaken).
 A: I assume that you mean "topological" in this sense, probably with the strict condition that $U(T)=X$ and $g=id$.  In that case, an almost-answer is:

$U$ is a bifibration in preorders.

A topological functor certainly has this property.  Conversely, if $U$ is a fibre-complete fibration in preorders, then given any $U$-structured source $(f_i : X \to U(S_i))$, we can form the family of reindexings $f_i^* S_i$ using the fibration structure and take their meet in the complete preorder $U^{-1}(X)$.  The initiality of this lift follows from the universal properties of cartesian arrows and meets, together with the fact that reindexing functors $g^*$ preserve meets, which follows from the fact that they have left adjoints because $U$ is also a bifibration.
Whether or not this is quite an answer depends on your definition of "fibre-complete" and size questions.  In order for the above argument to work, we need the fibers to have possibly large meets; a complete small preorder automatically has large meets, but in general the fibers of a topological functor may be large.  However if you define "fibre-complete" to mean the existence of all meets including large ones, then it works.
A: If U is topological is equivalent to have:


*

*U is faithful

*U admits lifting of one source and one sink

*U is fibre complete


You can see this  in 1, pp. 377, exercise 21C. also you can check  http://revistas.unal.edu.co/index.php/bolma/article/view/18271/19185 
