Example 3.4.1.4. Suppose we are given a commutative diagram of simplicial sets

3.41

\begin{equation} \label{diagram:homotopy-pullback-square2} \begin{gathered} \xymatrix { Y \ar [r] \ar [d]^{g} & X \ar [d]^{f} \\ T \ar [r] & S, } \end{gathered}\end{equation}

where $f$ and $g$ are Kan fibrations. Then (3.41) is homotopy Cartesian if and only if, for each vertex $t \in T$ having image $s \in S$, the induced map $Y_{t} \rightarrow X_{s}$ is a weak homotopy equivalence. This is essentially a reformulation of Proposition 3.3.7.1 (by virtue of Proposition 3.4.1.2).