  We provide a transparent, simple and unified treatment of recent results on
the equality conditions for the data processing inequality (DPI) of the
sandwiched quantum R\'enyi divergence, including the statement that equality in
the data processing implies recoverability via the Petz recovery map for the
full range of $\alpha$ recently proven by Jen\v cov\'a. We also obtain a new
set of equality conditions, generalizing a previous result by Leditzky et al.
