Answer
$T_0=T_L$
Work Step by Step
$$
\eta_{\mathrm{II}, \mathrm{HP}}=\frac{\dot{E} x_{\dot{Q}_H}}{\dot{W}}=\frac{\dot{W_{\min }}}{\dot{W}}=1-\frac{\dot{E} x_{\text {dest,total }}}{\dot{W}}
$$ Substituting $$
\dot{W}=\frac{\dot{Q}_H}{\mathrm{COP}_{\mathrm{HP}}} \quad \text { and } \quad \dot{E} x_{\dot{Q}_H}=\dot{Q}_H\left(1-\frac{T_0}{T_H}\right)
$$ The second-law efficiency equation $$
\eta_{\mathrm{II}, \mathrm{HP}}=\frac{\dot{E} \dot{x}_{\dot{Q}_H}}{\dot{W}}=\frac{\dot{Q}_H\left(1-\frac{T_0}{T_H}\right)}{\frac{\dot{Q}_H}{\mathrm{COP}_{\mathrm{HP}}}}=\dot{Q}_H\left(1-\frac{T_0}{T_H}\right) \frac{\mathrm{COP}_{\mathrm{HP}}}{\dot{Q}_H}=\frac{\mathrm{COP}_{\mathrm{HP}}}{\frac{T_H}{T_H-T_L}}=\frac{\mathrm{COP}_{\mathrm{HP}}}{\mathrm{COP}_{\text {Camot }}}
$$ since $T_0=T_L$.
