  It is easy to show coincidence of the entanglement-assisted classical
capacity and the Holevo capacity for any c-q channel between finite dimensional
quantum systems. In this paper we prove the converse assertion: coincidence of
the above-mentioned capacities of a quantum channel implies that the
$\chi$-essential part of this channel is a c-q channel (the $\chi$-essential
part is a restriction of a channel obtained by discarding all input states
useless for transmission of classical information).
  The above observations are generalized to infinite dimensional quantum
channels with linear constraints by using the obtained conditions for
coincidence of the quantum mutual information and the constrained Holevo
capacity.
