Definition 8

If A is a set, the set $\Delta_\mathrm{A}$ of all objects of the form $(x, x)$, where $x \in \mathrm{A}$, is called the diagonal of $\mathrm{A} \times \mathrm{A}$.

Clearly we have $\mathrm{pr}_1\Delta_\mathrm{A} = \mathrm{pr}_2\Delta_\mathrm{A} = \mathrm{A}$. The correspondence

is called the identity correspondence on A; it is its own inverse.

¶ If $\Gamma$ is a correspondence between A and B and if $\mathrm{I}_\mathrm{A}$, $\mathrm{I}_\mathrm{B}$ are the identity correspondences on A, B, respectively, then $\Gamma \circ \mathrm{I}_\mathrm{A} = \mathrm{I}_\mathrm{B} \circ \Gamma = \Gamma$.