One source of coalgebras in theoretical computer science begins with the idea that
coalgebras are generalized transition systems, with the final coalgebra
carried by all possible behaviors of all possible transition systems.   As such, the
final coalgebra may be taken as the 'codomain of semantics', with the domain
a set of terms of some kind or other.  
This leads to a research program of constructing languages for interesting
'transition trajectories'; typically the terms of these languages are finite objects
but the behaviors are infinite.    Part of this program involves logics for the equivalence
of terms.  This is where the logic of recursion (first proposed by Yiannis Moschovakis)
enters, since it is a general logic of fixed point terms.