  We extend Barr's well-known characterization of the final coalgebra of a
$Set$-endofunctor as the completion of its initial algebra to the
Eilenberg-Moore category of algebras for a $Set$-monad $\mathbf{M}$ for
functors arising as liftings. As an application we introduce the notion of
commuting pair of endofunctors with respect to the monad $\mathbf{M}$ and show
that under reasonable assumptions, the final coalgebra of one of the
endofunctors involved can be obtained as the free algebra generated by the
initial algebra of the other endofunctor.
