  A fraction of the mid-$J$ ($J$= 14--13 to $J$= 24--23) CO emission detected
by the \textit{Herschel}/PACS observations of embedded young stellar objects
(YSOs) has been attributed to the UV-heated outflow cavity walls. We have
applied our newly developed self-consistent models of Photon-Dominated Region
(PDR) and non-local thermal equilibrium line Radiative transfer In general Grid
(RIG) to the \textit{Herschel} FIR observations of 27 low mass YSOs and one
intermediate mass YSO, NGC7129-FIRS2. When the contribution of the hot
component (traced by transitions of $J> 24$) is removed, the rotational
temperature of the warm component is nearly constant with $\sim250$ K. This can
be reproduced by the outflow cavity wall ($n \geq 10^6\, \mathrm{cm}^{-3}$,
$\log G_{0}/n \geq-4.5$, $\mathrm{log} G_0\ge 3$, $T_{\rm gas} \ge 300 $K, and
X(CO)$ \ge 10^{-5}$) heated by a UV radiation field with a black body
temperature of 15,000 K or 10,000 K. However, a shock model combined with an
internal PDR will be required to determine the quantitative contribution of a
PDR relative to a shock to the mid-$J$ CO emission.
