  The classical powerset construction is a standard method converting a
nondeterministic automaton into a deterministic one recognising the same
language. Recently, the powerset construction has been lifted to a more general
framework that converts an automaton with side-effects, given by a monad, into
a deterministic automaton accepting the same language. The resulting automaton
has additional algebraic properties, both in the state space and transition
structure, inherited from the monad. In this paper, we study the reverse
construction and present a framework in which a deterministic automaton with
additional algebraic structure over a given monad can be converted into an
equivalent succinct automaton with side-effects. Apart from recovering examples
from the literature, such as the canonical residual finite-state automaton and
the \'atomaton, we discover a new canonical automaton for a regular language by
relating the free vector space monad over the two element field to the
neighbourhood monad. Finally, we show that every regular language satisfying a
suitable property parametric in two monads admits a size-minimal succinct
acceptor.
