  We present a systematical study on the prompt $J/\psi$ production in
association with a $c\bar{c}$ pair via the process, $\gamma\gamma \to
H(c\bar{c}) +c+\bar{c}$, within the framework of non-relativistic QCD at the
future high-energy $e^{+}e^{-}$ collider - International Linear Collider (ILC),
including both direct and feed-down contributions. For direct $J/\psi$
production, the states with color-octet channels, especially the $^3P_J^{[8]}$
and $^1S_0^{[8]}$ ones, provide dominant contribution to the production
cross-section, which are about fifty-two times over that of the color-singlet
one. This is clearly shown by the transverse momentum ($p_t$) and rapidity
distributions. The feed-down contribution from $\psi'$ and $\chi_{cJ}$
($J=0,1,2$) is sizable, which is about $20\%$ to the total prompt
cross-section. Besides the yields, we also calculate the $J/\psi$ polarization
parameter $\lambda$. In small $p_t$ region, the polarization of the prompt
$J/\psi$ is longitudinal due to large contributions through the $^3P_J^{[8]}$
channel, and becomes transverse in high $p_t$ region due to $^3S_1^{[8]}$
channel. Thus the $J/\psi$ production via photon-photon collisions at the ILC
shall provide a useful platform for testing the color-octet mechanism.
