Identity Arrow

Identity arrows are designated arrows ${\mathrm{id}}_x$ for all objects $x$ in a category such that.
$f\circ id=f$ and $\mathrm{id}\circ g=g$ for all suitable $f,g$.

Remarks

They are unique: ${\mathrm{id}}_x={\mathrm{id}}_x\circ {\mathrm{i}{\mathrm{d}}^{\prime }}_x={\mathrm{i}{\mathrm{d}}^{\prime }}_x$.