Remark 4.8.6.16 (Base Change). Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@C =50pt@R=50pt{ \operatorname{\mathcal{C}}' \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d] \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}, } \]
where the vertical maps are inner fibrations. Then, for every integer $n$, the simplicial set $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}'/\operatorname{\mathcal{D}}')}}$ can be identified with the fiber product $\operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} {\, }\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$. In particular, for every vertex $D \in \operatorname{\mathcal{D}}$, we have a canonical isomorphism
\[ \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \simeq \operatorname {h}_{\mathit{\leq {}n}}{\mathit{( \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}})}}. \]