  We present a complex-valued electric field model for experimentally observed
cavity transmission in coherent cavity-enhanced (CE) multiplexed spectroscopy
(i.e., dual-comb spectroscopy, DCS). The transmission model for CE-DCS differs
from that previously derived for Fourier-transform CE direct frequency comb
spectroscopy [Foltynowicz et al., Appl. Phys. B 110, 163-175 (2013)] by the
treatment of the local oscillator which, in the case of CE-DCS, does not
interact with the enhancement cavity. Validation is performed by measurements
of complex-valued near-infrared spectra of CO and CO$_2$ by an electro-optic
frequency comb coherently coupled to an enhancement cavity of finesse
$F=19600$. Following validation, we measure the $30012\leftarrow00001$
$^{12}$C$^{16}$O$_2$ vibrational band origin with a combined standard
uncertainty of 770 kHz (fractional uncertainty of $4\times10^{-9}$).
