Shoukanjuu

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Shoukanjuu (summoned beings) are avatars of the students of Fumizuki Academy. They appear as chibi versions of their students (with beast-like features) equipped with their own clothing and a weapon. The base attack power, defense power and health of a shokanjuu depends on the intelligence of a student, meaning that if you're smart, your shokanjuu's
battle statistics are high, while if you're idiotic they are not good. Akihisa's shokanjuu is the special one because he is the Kansatsu Shobunsha (Punishment Inspector) and his shokanjuu can touch physical objects, is very fast, and physically strong. However, any damage his shokanjuu receives will be transmitted as the equivalent pain to himself. A shokanjuu can either increase or decrease its attack and defense power. It depends on the score the student got on their latest exam. The students can replenish the health of their shokanjuu by taking a supplementary exam (or a "Recovery Test". This counts as their last test and will be used for the next battle.). If a shokanjuu loses in an Exam Summoning Battle, the student must take supplementary lessons with Ironman (except for Class F who has to take it every weekend, even though their shoukanjuu are untouched), to revive their shokanjuu. Some teachers have a shokanjuu of their own (i.e. Youko Takahashi) which can be corporeal to deal with students.

Source :

bakatotest.wikia.com/wiki/Baka_to_Test_to_Shoukanjuu_Wiki