  A unified $\chi$-criterion for heat devices (including heat engines and
refrigerators) which is defined as the product of the energy conversion
efficiency and the heat absorbed per unit time by the working substance [de
Tom\'{a}s \emph{et al} 2012 \textit{Phys. Rev. E} \textbf{85} 010104(R)] is
optimized for tight-coupling heat engines and refrigerators operating between
two heat baths at temperatures $T_c$ and $T_h(>T_c)$. By taking a new
convention on the thermodynamic flux related to the heat transfer between two
baths, we find that for a refrigerator tightly and symmetrically coupled with
two heat baths, the coefficient of performance (i.e., the energy conversion
efficiency of refrigerators) at maximum $\chi$ asymptotically approaches to
$\sqrt{\varepsilon_C}$ when the relative temperature difference between two
heat baths $\varepsilon_C^{-1}\equiv (T_h-T_c)/T_c$ is sufficiently small.
Correspondingly, the efficiency at maximum $\chi$ (equivalent to maximum power)
for a heat engine tightly and symmetrically coupled with two heat baths is
proved to be $\eta_C/2+\eta_C^2/8$ up to the second order term of $\eta_C\equiv
(T_h-T_c)/T_h$, which reverts to the universal efficiency at maximum power for
tight-coupling heat engines operating between two heat baths at small
temperature difference in the presence of left-right symmetry [Esposito
\emph{et al} 2009 \textit{Phys. Rev. Lett.} \textbf{102} 130602].
