  A long-lived magnetar, potentially originating from a binary neutron star
system, has been proposed to explain the extended emission observed in certain
short-duration gamma-ray bursts (sGRBs), and is posited as a potential central
engine to power the engine-fed kilonovae. Previously, the process by which
energy is injected into the surrounding ejecta/jet was widely believed to be
nearly isotropic. In this study, we employ special relativity
magnetohydrodynamic (SRMHD) simulations to investigate the wind injection
process from a magnetar central engine. We explore the dynamics and energy
distribution within the system and found that the parameter $\alpha=u_{\rm
A}/u_{\rm MWN}$ can be used to indicate the collimation of the magnetar wind
energy injection, where $u_{\rm A}$ is the local Alfven four-speed and $u_{\rm
MWN}$ is the four-speed of the magnetar wind nebular (MWN) formed from
wind-ejecta collision. A significant portion of the injected energy from the
magnetar spin-down wind will be channeled to the jet axis due to collimation
within the MWN. Achieving isotropic energy injection requires a significantly
small $\alpha$ that necessitates either an ultra-relativistic expanding MWN or
an extremely low magnetization MWN, both of which are challenging to attain in
sGRBs. Consequently, a considerably reduced energy budget (i.e. energy per
solid angle reduced by a factor of up to 10 with respect to the value under
isotropic assumption) is anticipated to be injected into the ejecta for
engine-fed kilonovae. Engine-fed kilonovae would appear fainter than originally
anticipated.
