  We investigate the $\sin (\phi_\Lambda-\phi_S)$ single-spin asymmetry in the
transversely polarized $\Lambda$ production in semi-inclusive deeply inelastic
scattering process within the framework of the transverse momentum
dependent~(TMD) factorization. The asymmetry is contributed by the convolution
of the polarizing TMD fragmentation function $D_{1T}^\perp$ of the $\Lambda$
hyperon and the unpolarized TMD distribution function $f_1$ of the target
nucleon. We apply two sets of $D_{1T}^{\perp,\Lambda^\uparrow/q}$, the
spectator diquark model result and available parametrization, to numerically
estimate the $\sin (\phi_\Lambda-\phi_S)$ asymmetry at the kinematical region
of electron ion collider (EIC). We take into account the TMD evolution effect
in order to include the scale dependence of the TMD distribution functions and
fragmentation 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 on
the $\sin (\phi_\Lambda-\phi_S)$ asymmetry with high precision at EIC can
provide important cross check on the available $\Lambda$ polarizing
fragmentation functions as well as constrain them more stringently.
