  We consider compound as well as arbitrarily varying classical-quantum channel
models. For classical-quantum compound channels, we give an elementary proof of
the direct part of the coding theorem. A weak converse under average error
criterion to this statement is also established. We use this result together
with the robustification and elimination technique developed by Ahlswede in
order to give an alternative proof of the direct part of the coding theorem for
a finite classical-quantum arbitrarily varying channels with the criterion of
success being average error probability. Moreover we provide a proof of the
strong converse to the random coding capacity in this setting.The notion of
symmetrizability for the maximal error probability is defined and it is shown
to be both necessary and sufficient for the capacity for message transmission
with maximal error probability criterion to equal zero. Finally, it is shown
that the connection between zero-error capacity and certain arbitrarily varying
channels is, just like in the case of quantum channels, only partially valid
for classical-quantum channels.
