Let $\mathbb{C}$ be a small/locally small category and let $T:\mathbb{C} \to \mathbb{C}$ be an endofunctor. One can then have $T$-algebras and $T$-coalgebras in the usual way: for $X,Y \in \mathbb{C}$, $f_x: TX \to X$ and $f_y: Y \to TY$, we have the algebra $(X, f_x)$ and the coalgebra $(X, f_y)$.
Can anything interesting at all be said about such algebras/coalgebras in the degenerate case when $T$ is an identity functor in $\mathbb{C}$? All I could think of is the obvious fact that such algebra/coalgebra are trivially self dual (and obviously bialgebras for $T$).
Any pointers on references connecting algebras to coalgebras (apart from the apparent duality) would be appreciated!