 First, for any set S, the functor Fa = S×  a.     A coalgebra for this F is a stream system of equations as we saw it in  Section 1.1,  except that there we made things concrete and took S to be the set of natural numbers. The final coalgebra is the set S∞ =  S ×   S∞  of  streams over S.    The logical language for this functor would be a sentential  (propositional) language  whose sentences are either true  or of the form s : φ  where s∈S. The semantics would be the obvious one; for example 