  A celebrated result in representation theory is that of higher Auslander
correspondence. Let $\Lambda$ an Artin algebra and $X$ a $d$-cluster-tilting
module. Iyama has shown that the endomorphism ring $\Gamma$ of $X$ is a
$d$-Auslander algebra, and moreover this gives a correspondence between
$d$-cluster-tilting modules and $d$-Auslander algebras. We present a
self-contained and concise proof using the homological theory of idempotent
ideals of Auslander--Platzeck--Todorov.
