Kerodon

Proposition 4.7.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. An object $X \in \operatorname{\mathcal{C}}$ is final if and only if, for every integer $m \geq 1$ and every morphism of simplicial sets $\sigma : \operatorname{\partial \Delta }^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\sigma (m) = X$, there exists an $m$-simplex $\overline{\sigma }: \Delta ^ m \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{\sigma }|_{ \operatorname{\partial \Delta }^ m} = \sigma $.