Kerodon

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

Example 2.1.1.4 (Endomorphism Categories). Let $\operatorname{\mathcal{C}}$ be a category, and let $\operatorname{End}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}})$ denote the category of functors from $\operatorname{\mathcal{C}}$ to itself. Then the composition functor

\[ \circ : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \times \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \mapsto \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{C}}) \quad \quad (F,G) \mapsto F \circ G; \]

is a nonunital strict monoidal structure on $\operatorname{End}(\operatorname{\mathcal{C}})$.