Definition 7

Let $\Gamma = (\mathrm{G}, \mathrm{A}, \mathrm{B})$ and $\Gamma' = (\mathrm{G}', \mathrm{B}, \mathrm{C})$ be two correspondences such that the target of $\Gamma$ is the source of $\Gamma'$. Then the correspondence $(\mathrm{G}' \circ \mathrm{G}, \mathrm{A}, \mathrm{C})$ is called the composition of $\Gamma'$ and $\Gamma$, and is denoted by $\Gamma' \circ \Gamma$ (or sometimes $\Gamma'\Gamma$).

It follows immediately from Proposition 5 that if X is a subset of A we have $(\Gamma' \circ \Gamma)\langle \mathrm{X}\rangle = \Gamma'\langle \Gamma\langle \mathrm{X}\rangle\rangle$. Furthermore, since the target of $\overset{-1}{\Gamma'}$ is the same as the source of $\overset{-1}{\Gamma}$, the inverse of $\Gamma' \circ \Gamma$ is $\overset{-1}{\Gamma} \circ \overset{-1}{\Gamma'}$, by Proposition 3.