We give two generic proofs for cut elimination in propositional modal
logics, interpreted over coalgebras. We first investigate semantic coherence
conditions between the axiomatisation of a particular logic and
its coalgebraic semantics that guarantee that the cut-rule is admissible
in the ensuing sequent calculus. We then independently isolate a
purely syntactic property of the set of modal rules that guarantees cut
elimination. Apart from the fact that cut elimination holds, our main
result is that the syntactic and semantic assumptions are equivalent in
case the logic is amenable to coalgebraic semantics. As applications
we present a new proof of the (already known) interpolation property
for coalition logic and newly establish the interpolation property for
the conditional logics CK and CK + ID.