Definition:Identity Morphism

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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.

Also see