# Kerodon

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Remark 2.2.0.3. To every strict $2$-category $\operatorname{\mathcal{C}}$, we can associate an ordinary category $\operatorname{\mathcal{C}}_0$, whose objects and morphisms are given by

$\operatorname{Ob}(\operatorname{\mathcal{C}}_0) = \operatorname{Ob}(\operatorname{\mathcal{C}}) \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( X, Y) = \operatorname{Ob}( \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) ).$

We will refer to $\operatorname{\mathcal{C}}_0$ as the underlying ordinary category of $\operatorname{\mathcal{C}}$ (note that $\operatorname{\mathcal{C}}_0$ can be obtained from $\operatorname{\mathcal{C}}$ by the general procedure of Example 2.1.7.5). More informally, the underlying category $\operatorname{\mathcal{C}}_0$ is obtained from $\operatorname{\mathcal{C}}$ by “forgetting” its $2$-morphisms.