Remark 2.4.2.11. Let $\operatorname{\mathcal{C}}$ be a strict $2$-category. Then the simplicial category $\operatorname{\mathcal{C}}_{\bullet }$ of Example 2.4.2.8 is locally Kan if and only if every $2$-morphism in $\operatorname{\mathcal{C}}$ is invertible: that is, if and only if $\operatorname{\mathcal{C}}$ is a $(2,1)$-category (in the sense of Definition 2.2.8.5). This follows from Proposition 1.3.5.2.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$