In vert kicking, the wrap term has a number of different meanings and uses. In general, as a takeoff, it is a cheat takeoff performed from backside, or one starting in frontside that continues past the usual, accepted takeoff point (over-cheating). Some would argue this is the function of the “K-step” takeoff.
As a transition, wraps, or wrapthroughs, act like a swingthrough when landing on the inside leg, almost exclusively from a hypered round rather than a hook kick. In this scenario, the performer lands in approximately backside on their inside leg. As they land, the chambered outside leg continues around, and even backward, initiating takeoff. Some people simply try to punch off of the leg as hard as possible. For this reason it can be argued to be a number of different transitions, although the wrap tag is by far the most convenient because it is so widely understood in this context.
in the clip above, a cheat-360-katana w/t 720-hook is performed. (540 w/t 1080)