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Remark 2.2.0.7 (Endomorphism Categories). Let $\operatorname{\mathcal{C}}$ be a strict $2$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We will write $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ for the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$. Then the composition law

\[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \]

determines a strict monoidal structure on the category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$.

Note that, if $\operatorname{\mathcal{C}}$ is an ordinary category (regarded as a strict $2$-category by means of Example 2.2.0.6), then the endomorphism category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ can be identified with the endomorphism monoid $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ of Example 1.3.2.2, regarded as a (strict) monoidal category via Example 2.1.2.8.