Kirchhoff’s Law states that for a blackbody emissivity = absorptivity. In our wave equation with small damping as a model of blackbody in radiative equilibrium with an exterior forcing, this is built into the model for frequencies below high-frequency cut-off.
This is because for these frequencies the outgoing radiation simply reproduces the part of the incoming radiation which is not reflected. Or put differently: In radiative equilibrium there is below cut-off no distinction between incoming radiation and outgoing radiation as explained in the previous post.
However, if the incoming waves (the forcing) contain frequencies above the present cut-off of the absorbing body, then the game changes, as these frequencies contribute to the internal energy of the body increasing its temperature and are not (re-)emitted. This is a case with positive absorptivity and zero emissivity of the frequencies above cut-off.
This leads to coefficients of absorptivity depending on temperature, which complicates the picture.
The effect is that different bodies with different cut-off at the same distance to the Sun, can take on different equilibrium temperatures, as evidenced in e.g. Satellite Thermal Control Engineering and What Is the Temperature of Space?