Thomason’s homotopy colimit theorem

R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Camb. Phil. Soc. (1) 85 (1979)91-109

says that for a **functor** $F:I^{op}\to Cat$, the homotopy colimit and pseudo-colimit are the same. More precisely, there exists a natural homotopy equivalence
$$
hocolimt(N F)\to N(\int F),
$$
where $\int F$ is the Grothendieck construction (computes pseudo-colimit).

My question is, what if we consider a **pseudo-functor** $F: I^{op}\to Cat$.

Now the right hand side is still defined. For the left side, he says it is not defined. But if we compose $F$ with nerve, we obtain a pseudo-functor $F: I^{op}\to SSet$, which induces a functor $hoNF: I^{op}\to hoSSet$, then take colimit in $hoSSet$. Are the two sides homotopical equivalent?