Identity morphism

Definition

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:

• For every morphism $f : X \to Y$, one has $f \circ 1_X = f$.
• For every morphism $g : Y \to X$, one has $1_X \circ g = g$.

(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

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