Kerodon

Remark 3.3.7.3. Let $f: X \rightarrow S$ be a Kan fibration of simplicial sets. If $s$ and $t$ are vertices of $S$ which belong to the same connected component, then the Kan complexes $X_{s}$ and $X_{t}$ are homotopy equivalent. To prove this, we may assume without loss of generality that there is an edge of $S$ with source $s$ and target $t$. Replacing $f$ by the projection map $\Delta ^1 \times _{S} X \rightarrow \Delta ^1$, we are reduced to the case where $S = \Delta ^1$; in this case, the Example 3.3.7.2 guarantees that the inclusion maps $X_{s} \hookrightarrow X \hookleftarrow X_{t}$ are weak homotopy equivalences.