  The detection of the short gamma-ray burst (SGRB) 050709 by the HETE-2
satellite opened a new window into understanding the nature of SGRBs, offering
clues about their emission mechanism and progenitors, with the crucial aid of
optical follow-up observations. Here, we revisit the prompt emission of GRB
050709. Our analysis reveals an initial hard spike ~200 ms long, followed by a
subsequent soft tail emission lasting ~300 ms. These components could be common
among other SGRBs originating from binary neutron merger events, such as GW/GRB
170817A. Detailed temporal and spectral analyses indicate that the soft tail
emission might be attributed to the cocoon formed by the relativistic jet
depositing energy into the surrounding material. We find the necessary cocoon
parameters at the breakout, as consistent with numerical simulation results. We
compared the physical parameters of this cocoon with those of other SGRBs. The
relatively higher cocoon pressure and temperature in GRB 050709 may indicate a
more on-axis jet compared to GRB 170817A and GRB 150101B.
