  The free algebra adjunction, between the category of algebras of a monad and
the underlying category, induces a comonad on the category of algebras. The
coalgebras of this comonad are the topic of study in this paper (following
earlier work). It is illustrated how such coalgebras-on-algebras can be
understood as bases, decomposing each element x into primitives elements from
which x can be reconstructed via the operations of the algebra. This holds in
particular for the free vector space monad, but also for other monads, like
powerset or distribution. For instance, continuous dcpos or stably continuous
frames, where each element is the join of the elements way below it, can be
described as such coalgebras. Further, it is shown how these
coalgebras-on-algebras give rise to a comonoid structure for copy and delete,
and thus to diagonalisation of endomaps like in linear algebra.
