Here is a theorem from Weibel's The K-book, Chapter IV
Theorem 3.6.1. Let f : X → Y be a map of bisimplicial sets.
(i) If each simplicial map $X_{p,∗} → Y_{p,∗}$ is a homotopy equivalence, so is BX → BY .
(ii) If Y is the nerve of a category I (constant in the second simplicial coordinate), and $f^{−1}(i,_\bullet) → f^{−1}(j,_\bullet)$ is a homotopy equivalence for every i → j in I, then each $B(f^{−1}(i)) → BX → B(I)$ is a homotopy fibration sequence.
In the above statements what is the map $f^{-1}(i,_\bullet)\to f^{-1}(j,_\bullet)$ induced by $i\to j$?
I am assuming that $f^{-1}(i,_\bullet)$ is the pullback of the map $f:X\to Y$ along the map $\Delta^0\to Y$ corresponding to $i$.