Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category object in sheaves of sets. By taking the nerve, one can view $F$ as a simplicial sheaf. $F$ will not take values in Kan complexes however (unless $F$ takes values in groupoids).
Question: Are there checkable conditions for $F$ to satisfy homotopy descent (besides the Cech diagram consisting of fibrations), in the sense that if $RF$ is the fibrant replacement of $F$ in the Joyal model structure on simplicial sheaves (so I'm modelling $\infty$-sheaves of $\infty$-groupoids here), then the canonical map $$F \to RF$$ is object-wise a weak equivalence of simplicial sets?