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We present a phenomenological study on the role of charm contribution and$SU(2)$ isospin symmetry in the extraction of the $\Lambda$ polarizingfragmentation functions from $e^+e^- \to \Lambda^\uparrow(\bar\Lambda^\uparrow) \,h + X$ annihilation processes. We adopt thewell-established transverse-momentum-dependent factorization formalism, withinthe Collins-Soper-Sterman evolution scheme at next-to-leading logarithmaccuracy, carefully exploiting the role of the nonperturbative component of thepolarizing fragmentation function. We then discuss the impact of these resultson the predictions for transverse $\Lambda$, $\bar{\Lambda}$ polarization insemi-inclusive deep inelastic scattering processes at typical energies of thefuture Electron-Ion Collider.

Transverse $\Lambda$ polarization in $e^+e^-$ annihilations and in SIDIS processes at the EIC within TMD factorization