I take the definition of a topological category from Segal and (later) Moerdijk. So this is a small category with topologies on the set of objects and the set of morphisms, such that the domain map, the codomain map, the identity map and the composition map are continuous.
Moreover, one can take the classifying topos $\mathcal B\mathbf C$ of such a category, as defined in Moerdjik, Classifying spaces and classifying topos. This is actually a Grothendieck topos, but who is a site of definition? Moerdijk says that one can prove that this is a topos by checking the Giraud axioms, but I would rather see a direct evidence, so a site $D$ and a Grothendieck topology s.t. $\mathcal B\mathbf C\cong Sh(D,J)$.
So in brief: in what sense is a topological category (e.g. a topological groupoid) a site in the sense of Grothendieck topologies?
Thank you in advance.