Remark 4.8.8.13 (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 $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}'/\operatorname{\mathcal{D}}')}$ can be identified with the fiber product $\operatorname{\mathcal{D}}' \times _{\operatorname{\mathcal{D}}} {\, }\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$. In particular, for every vertex $D \in \operatorname{\mathcal{D}}$, we have a canonical isomorphism
\[ \{ D\} \times _{\operatorname{\mathcal{D}}} \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \simeq \mathrm{h}_{\mathit{\leq n}}\mathit{( \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}})}. \]