\begin{definition}[Definition:Identity Morphism]
Let $\mathbf C$ be a metacategory.
Let $X$ be an object of $\mathbf C$.
The '''identity morphism''' of $X$, denoted $\operatorname{id}_X$, is a morphism of $\mathbf C$ subject to:
:$\operatorname{dom} \operatorname{id}_X = \operatorname{cod} \operatorname{id}_X = X$
:$f \circ \operatorname{id}_X = f$
:$\operatorname{id}_X \circ g = g$
whenever $X$ is the domain of $f$ or the codomain of $g$, respectively.
In most metacategories, the '''identity morphisms''' can be viewed as a representation of "doing nothing", in a sense suitable to the metacategory under consideration.
\end{definition}