  This paper focus on the presheaf monad and its submonads on the realm of
$V$-categories, for a quantale $V$. First we present two characterisations of
presheaf submonads, both using $V$-distributors: one based on admissible
classes of $V$-distributors, and other using Beck-Chevalley conditions on
$V$-distributors. Then we focus on the study of the corresponding
Eilenberg-Moore categories of algebras, having as main examples the formal ball
monad and the so-called Lawvere monad.
