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We investigate the $\sin (\phi_\Lambda-\phi_S)$ single-spin asymmetry in thetransversely polarized $\Lambda$ production in semi-inclusive deeply inelasticscattering process within the framework of the transverse momentumdependent~(TMD) factorization. The asymmetry is contributed by the convolutionof the polarizing TMD fragmentation function $D_{1T}^\perp$ of the $\Lambda$hyperon and the unpolarized TMD distribution function $f_1$ of the targetnucleon. We apply two sets of $D_{1T}^{\perp,\Lambda^\uparrow/q}$, thespectator diquark model result and available parametrization, to numericallyestimate the $\sin (\phi_\Lambda-\phi_S)$ asymmetry at the kinematical regionof electron ion collider (EIC). We take into account the TMD evolution effectin order to include the scale dependence of the TMD distribution functions andfragmentation functions. It is found that different sets of$D_{1T}^{\perp,\Lambda^\uparrow/q}$ lead to different ${\sin(\phi_\Lambda-\phi_S)}$ asymmetry, particularly in sign. Future measurements onthe $\sin (\phi_\Lambda-\phi_S)$ asymmetry with high precision at EIC canprovide important cross check on the available $\Lambda$ polarizingfragmentation functions as well as constrain them more stringently.

$\sin (\phi_\Lambda-\phi_S)$ azimuthal asymmetry in transversely polarized $\Lambda$ production in SIDIS within TMD factorization at EIC