Example 1.3.2.2. Let $\operatorname{\mathcal{C}}$ be a category and let $X$ be an object of $\operatorname{\mathcal{C}}$. An endomorphism of $X$ is a morphism from $X$ to itself in the category $\operatorname{\mathcal{C}}$. We let $\operatorname{End}_{\operatorname{\mathcal{C}}}(X) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$ denote the set of all endomorphisms of $X$. The composition law on $\operatorname{\mathcal{C}}$ determines a map
which exhibits $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as a monoid; the unit element of $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ is the identity morphism $\operatorname{id}_{X}: X \rightarrow X$. We refer to $\operatorname{End}_{\operatorname{\mathcal{C}}}(X)$ as the endomorphism monoid of $X$.