  We investigate the transition monoid construction for deterministic automata
in a categorical setting and establish it as an adjunction. We pair this
adjunction with two other adjunctions to obtain two endofunctors on
deterministic automata, a comonad and a monad, which are closely related,
respectively, to the largest set of equations and the smallest set of
coequations satisfied by an automaton. Furthermore, we give similar transition
algebra constructions for lasso and {\Omega}-automata, and show that they form
adjunctions. We present some initial results on sets of equations and
coequations for lasso automata.
