# Kerodon

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Example 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be a $(2,1)$-category, so that the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ is an $\infty$-category (Theorem 2.3.2.1). Combining Remark 4.6.6.6) with Example 4.6.5.12, we obtain the following:

• An object $Y \in \operatorname{\mathcal{C}}$ is initial when viewed as an object of the $\infty$-category $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ if and only if, for every object $Z \in \operatorname{\mathcal{C}}$, the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$ is contractible (that is, there exists a $1$-morphism from $Y$ to $Z$ and for every pair of morphisms $f,g: X \rightarrow Y$, there is a unique isomorphism $\gamma : f \xRightarrow {\sim } g$).

• An object $Y \in \operatorname{\mathcal{C}}$ is final when viewed as an object of the $\infty$-category $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ if and only if, for every object $X \in \operatorname{\mathcal{C}}$, the groupoid $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ is contractible.