  We show that a monomial algebra $\Lambda$ over an algebraically closed field
$K$ is self-injective if and only if each map
$\mathrm{soc}(_{\Lambda}\Lambda)\to \ _{\Lambda}\Lambda$ can be extended to an
endomorphism of $_{\Lambda}\Lambda$, and provide a complete classification of
such algebras. As a consequence, we show that the class of self-injective
monomial algebras is a subclass of Nakayama algebras.
