  In this work we derive some general features of the redshift measured by
radially moving observers in the black hole background. Let observer 1 cross
the black hole horizon emitting of a photon while observer 2 crossing the same
horizon later receives it. We show that if (i) the horizon is the outer one
(event horizon) and (ii) it is nonextremal, received frequency is redshifted.
This generalizes previous recent results in literature. For the inner horizon
(like in the Reissner-Nordstr\"{o}m metric) the frequency is blueshifted. If
the horizon is extremal, the frequency does not change. We derive explicit
formulas describing the frequency shift in generalized Kruskal- and
Lemaitre-like coordinates.
