An algebra for a monad $T : \mathcal{C} \to \mathcal{C}$ is an arrow $m : Tc \to c$ satisfying some relevant conditions regarding the unit $\eta : 1 \Rightarrow T$ and the multiplication $\mu : T^2 \Rightarrow T$. For the free-forgetful adjunction of groups, $F \dashv U$, we get a monad $T = UF : \mathsf{Set} \to \mathsf{Set}$, and the algebras of this monad are exactly the groups.
We can dualize this and look at the comonad $FU : \mathsf{Grp} \to \mathsf{Grp}$. Coalgebras for this comonad are group homomorphisms $w : G \to FUG$ with conditions dual to those we placed on algebras. Most notably, we require the composite $G \overset{w}{\to} FUG \overset{\epsilon_G}{\to} G$ be the identity, so (unless I'm making a silly mistake) we are forced to let $w$ be the obvious inclusion of $G$ into $FUG$ (which is a homomorphism precisely when $G$ is free).
I can't find any discussions of the comonad associated to a free-forgetful adjunction, despite them being a go-to example of adjunctions in general. Moreover, all of the examples I can find regarding coalgebras are somehow infinitary in nature. E.g. coalgebras for the stream monad are exactly forward orbits of a dynamical system. Is there anything interesting to say about coalgebras of a free-forgetful adjunction? My previous paragraph seems to indicate "no", but I would love to be pleasantly surprised.
Thanks in advance ^_^