I have a problem concerning a fact which is stated without proof in this Rezk's draft: http://www.math.uiuc.edu/~rezk/i-hate-the-pi-star-kan-condition.pdf . In the proof of Lemma 2.11, we are given a fibration $p:E \to B$ and a diagram $$V: J \to Psh(I)/B$$ where $Psh(I)$ denotes simplicial presheaves on $I$, such that any arrow $V(j) \to B$ is such that the pullback of $p$ along it gives us a square (a homotopy cartesian one, of course) which remains homotopy cartesian after having applied the homotopy colimit functor $$hocolim:Psh(I)\to sSet$$ The claim is that the map $$hocolim(V)\to B$$ should then have the same property. What I have done so far is firstly having interpreted this claim as: pick a cofibrant replacement $\mathcal{Q}V\simeq V$ in $Psh(I)^J$ and consider the map $colim \mathcal{Q}V \to B$, we want it to have the abovementioned property (notice that the maps $\mathcal{Q}V(j) \to B$ have it, since they are weakly equivalent to the $V(j) \to B$ which have it by assumption).

I would like to use the following result, but I haven't find a way in which it could turn out to solve my problem:

Here, $E \to B$ is equifibered if the following square is homotopy cartesian for any choice of $J_1,J_2,\alpha$:

And $E \to B$ is a realization-fibration if any homotopy cartesian square of the form remains homotopy cartesian after having applied the homotopy colimit functor

Thanks in advance for any hint or advice.