**Tensor Hopf endofunctor**

**Abstract:**

An approach to A_{¥}-categories presented at the book "Closed multicategory of pretriangulated A_{¥}-categories" (joint with Volodymyr
Lyubashenko and Oleksandr Manzyuk) involves construction of Kleisli multicategory for free cocategory comonad T^{ ³ 1} on the Monoidal category of
augmented graded k-linear quivers.

The talk is devoted to an observation concerning Hopf algebra structure on the endofunctor T^{ ³ 1}. We explain how bialgebra projection appear
in the construction of A_{¥}-analog of Verdier factor-category. Finally, we discus some new constructions from higher dimensional algebra that concern
the matter.