  Lax extensions of set functors play a key role in various areas including
topology, concurrent systems, and modal logic, while predicate liftings provide
a generic semantics of modal operators. We take a fresh look at the connection
between lax extensions and predicate liftings from the point of view of
quantale-enriched relations. Using this perspective, we show in particular that
various fundamental concepts and results arise naturally and their proofs
become very elementary. Ultimately, we prove that every lax extension is
induced by a class of predicate liftings; we discuss several implications of
this result.
