Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 2.2.1.7 (Endomorphism Categories). Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We will denote the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ by $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ refer to it as the endomorphism category of $X$. The category $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ has a monoidal structure, with tensor product is given by the composition law

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

unit object given by the identity $1$-morphism $\operatorname{id}_{X}$, and the unit and associativity constraints of $\underline{\operatorname{End}}_{\operatorname{\mathcal{C}}}(X)$ given by $\upsilon _{X}$ and the associativity constraints of $\operatorname{\mathcal{C}}$, respectively.