Consider a small diagram $X_\bullet$ of simplicial sets. Under what sufficient conditions (apart from $X_\bullet$ being injective fibrant), is the map $\text{lim }X_\bullet \to \text{holim } X_\bullet$ a weak equivalence? Is it sufficient that the morphisms of simplicial sets $X_{i} \rightarrow X_{j}$ be fibrations $\forall i,j$?

I'm looking for something analogous to what one has in case of homotopy pullbacks of simplicial sets where one map being a fibration is sufficient to ensure that homotopy pullback is same as categorical pullback.

EDIT: I'm interested in diagrams indexed by cosimplex category $\Delta$.