In this paper, we study linear rewriting systems modulo a set of algebraic axioms. We introduce the structure (3,2)-polygraph as presentation category enriched in categories, (called (2,2)-category), by system symbolic computation method order to compute bases for vector spaces 2-cells these categories. particular, case pivotal 2-categories using isotopy relations given biadjunctions on 1-cells...

Given a pseudomonad $mathcal{T} $ on a $2$-category $mathfrak{B} $, if a right biadjoint $mathfrak{A}tomathfrak{B} $ has a lifting to the pseudoalgebras $mathfrak{A}tomathsf{Ps}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $ then this lifting is also right biadjoint provided that $mathfrak{A} $ has codescent objects. In this paper, we give  general results on lifting of biadjoints. As a consequence, ...