# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Proposition 5.4.1.4. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category.

Proof. Our assumption that $\operatorname{\mathcal{C}}$ is an $\infty$-category guarantees that every $2$-simplex of $\operatorname{\mathcal{C}}$ is thin (Example 2.3.2.4). Consequently, condition $(2)$ of Definition 5.4.1.3 is automatic, and condition $(1)$ follows immediately from the definition. Conditions $(3)$ and $(4)$ follow from Theorem 4.4.2.6 (since every degenerate edge of $\operatorname{\mathcal{C}}$ is an isomorphism). $\square$