Remark 10.2.2.30. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $Y_{\bullet }$ be a simplicial object of $\operatorname{\mathcal{C}}$, and suppose we are given a morphism $f_{\bullet }: X_{\bullet } \rightarrow Y_{\bullet }$ of semisimplicial objects of $\operatorname{\mathcal{C}}$. It follows from the proof of Proposition 10.2.2.29 that $f_{\bullet }$ exhibits $Y_{\bullet }$ as a left Kan extension of $X_{\bullet }$ along the inclusion map $\operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}_{\operatorname{inj}}^{\operatorname{op}} ) \subseteq \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )$ if and only if, for every integer $n \geq 0$, the following condition is satisfied:
- $(\ast _ n)$
Let $E$ be the collection of all surjections $\alpha : [n] \twoheadrightarrow [m]$ in the category $\operatorname{{\bf \Delta }}$. For each $\alpha \in E$, let $g_{\alpha }: X_{m} \rightarrow Y_{n}$ be a composition of $f_{m}: X_{m} \rightarrow Y_{m}$ with the morphism $\alpha ^{\ast }: Y_{m} \rightarrow Y_{n}$. Then the collection $\{ g_{\alpha } \} _{\alpha \in E}$ exhibit $Y_{n}$ as a coproduct of the collection of objects $\{ X_{m} \} _{ (\alpha : [n] \twoheadrightarrow [m]) \in E }$.
Compare with Construction 3.3.1.6.