Remark 4.8.8.15. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n$ be an integer. Then the comparison map $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ of Construction 4.8.8.10 fits into a commutative diagram
\[ \xymatrix@C =50pt@R=50pt{ & \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \ar [dr]^{G} & \\ \operatorname{\mathcal{C}}\ar [ur]^{F'} \ar [rr]^{F} & & \operatorname{\mathcal{D}}. } \]
For $n \geq 0$, the morphism $F'$ carries each $m$-simplex of $\operatorname{\mathcal{C}}$ to the $m$-simplex $( F(\sigma ), \tau )$ of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$, where $\tau $ is the composite map
\[ \Delta ^{m} \xrightarrow {(\operatorname{id},\sigma )} \Delta ^{m} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_{\sigma } \rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{( \operatorname{\mathcal{C}}_{\sigma } )}. \]