LNL polycategories and doctrines of linear logic

We define and study LNL polycategories, which abstract the judgmental structure of classical linear logic with exponentials. Many existing structures can be represented as including adjunctions, exponential comonads, multicategories, IL-indexed categories, linearly distributive categories storage, commutative strong monads, CBPV-structures, models polarized calculi, Freyd-categories, skew well ordinary cartesian, symmetric, planar multicategories monoidal symmetric *-autonomous categories. To such classes uniformly, we a notion doctrine, that each these identified algebras for some doctrine. show free doctrines presented by sequent calculus, every morphism induces an adjunction between their 2-categories algebras.