  The disk of an active galactic nucleus (AGN) is widely regarded as a
prominent formation channel of binary black hole (BBH) mergers that can be
detected through gravitational waves (GWs). Besides, the presence of dense
environmental gas offers the potential for an embedded BBH merger to produce
electromagnetic (EM) counterparts. In this paper, we investigate EM emission
powered by the kicked remnant of a BBH merger occurring within the AGN disk.
The remnant BH will launch a jet via accreting magnetized medium as it
traverses the disk. The resulting jet will decelerate and dissipate energy into
a lateral cocoon during its propagation. We explore three radiation mechanisms
of the jet cocoon system: jet breakout emission, disk cocoon cooling emission,
and jet cocoon cooling emission, and find that the jet cocoon cooling emission
is more likely to be detected in its own frequency bands. We predict a soft
X-ray transient, lasting for O($10^3$) s, to serve as an EM counterpart, of
which the time delay O(10) days after the GW trigger contributes to follow-up
observations. Consequently, BBH mergers in the AGN disk represent a novel
multimessenger source. In the future, enhanced precision in measuring and
localizing GWs, coupled with diligent searches for such associated EM signal,
will effectively validate or restrict the origin of BBH mergers in the AGN
disk.
