Let $\mathcal{C}$ be a category with hom-sets, and let $X$ be an object of $\mathcal{C}$. Define

\begin{align*} \operatorname{End}_{\mathcal{C}}(X) := \operatorname{Hom}_{\mathcal{C}}(X,X). \end{align*}

Then $\operatorname{End}_{\mathcal{C}}(X)$ is closed under categorical composition: if $f,g \in \operatorname{End}_{\mathcal{C}}(X)$, then $f \circ g \in \operatorname{End}_{\mathcal{C}}(X)$. Moreover, $\operatorname{End}_{\mathcal{C}}(X)$ is a monoid under composition, with identity element $\operatorname{id}_X$. Equivalently, for all $f,g,h \in \operatorname{End}_{\mathcal{C}}(X)$,

\begin{align*} (f \circ g) \circ h = f \circ (g \circ h), \end{align*}

and for every $f \in \operatorname{End}_{\mathcal{C}}(X)$,

\begin{align*} \operatorname{id}_X \circ f = f = f \circ \operatorname{id}_X. \end{align*}