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Taotie is the spirit that is used originally by the Rosenthal Family to posses corpses in their necromancy magic. When Esther Rosenthal was "tricked" into helping Hishigata into his research, the Taotie would eventually be used to possess the corpse of an esper that were to operate the mecha "body" of the Coffin.
Originally, the esper's teleport was very weak, as it could only swap out their position with their position with their favorite stuffed animal. The ability is later strengthened due to the Coffin acting as the esper's "body".
The Asport is an even further strengthened version of the teleport by visually linking Qiong Qi and Taotie, allowing Taotie to use its power beyond the scope of its own vision. While it only worked on the stuffed animal when it came to esper, and for the Taotie it was hypnotized into believing what the stuffed animal was when using its power, now anyone or anything can be forced into the role of the stuffed animal. It is somewhat similar to Musujime Awaki's Move Point, not needing to physically touch an object to teleport it, both in scale and the distance in which the Asport can use its powers, not even needing to be physically present or even near the target of its teleportation.
Hishigata used its teleportation powers alongside Qiong Qi's Level 5 Psychokinesis ability against Accelerator. It can teleport something as large as Qiong Qi, and indeed, it was powerful enough to bypass Accelerator's ability that ordinary teleport abilities cannot usually bypass, allowing Accelerator to be teleported without his consent. Despite this, the ability can only be used one at a time, as Taotie could not teleport Qiong Qi and Accelerator at the same time.