Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be an isofibration of small categories (fibration of categories in the canonical sense). Under what conditions is the nerve $N(F):N(\mathcal{C})\rightarrow N(\mathcal{D})$ a Kan fibration of simplicial sets?



When is an isofibration a Kan Fibration?
This follows from HTT, Prop 2.1.1.3. A map of categories is a Grothendieck fibration that is (co)fibered in groupoids if and only if the map on nerves is a left (right) Kan fibration, so by loc. cit. an isofibration induces a Kan fibration after taking nerves if and only if it's fibered and cofibered in groupoids, as Dylan says in the comments.


