  Experimental data from Belle Collaboration for the transverse polarization of
$\Lambda$'s measured in $e^+ e^-$ annihilation processes are used to extract
the polarizing fragmentation function (FF) within a TMD approach. We consider
both associated and inclusive $\Lambda$ production, showing a quite consistent
scenario. Good separation in flavor is obtained, leading to four independent
FFs. Predictions for SIDIS processes at the EIC, crucial for understanding
their universality and evolution properties, are also presented.
