 Examples of final coalgebras and corecursive definitions.  Our conceptual comparison makes the point that algebras embody  sets with operations.    This point is almost too easy:    the reason behind the terminology of  “algebras” in category theory is that sets with operations may be modeled as algebras in the categorical sense.    For coalgebras, it is harder to make the case that they directly correspond to sets with either “transitions” or “observations”. However, we present a few examples that motivate this point. 