Identity morphism - Knowlpedia

Let $\mathcal C$ be a category and let $X$ be an object of $\mathcal C$ . An identity morphism on $X$ is a morphism $1_X : X \to X$ such that:

(Here $\circ$ is composition.)

Uniqueness: If $e : X \to X$ also satisfies these unit laws, then $e = 1_X$ .

This generalizes the identity function on a set.

Examples

In $\mathbf{Set}$ , $1_X$ is the identity function $x\mapsto x$ on the set $X$ .
In $\mathbf{Grp}$ , $1_G : G\to G$ is the identity group homomorphism.
In $\mathbf{Top}$ , $1_X : X\to X$ is the identity continuous map on a space $X$ .