In the paper B. Bakalov, A.D'Andrea and V.G.Kac, Theory of finite pseudoalgebras, section 3, one finds the following definition of pseudotensor category:
A pseudotensor category is a class of objects $\mathcal {M}$ together with vector spaces Lin$(\{L_i, M\}),$ equipped with actions of symmetric groups $S_I$ among them and composition maps, satisfying Associativity, Unit, and Equivariance axioms.
My question is:
Why does the definition require the action of symmetric groups $S_I$ ?
How does the action of $S_I$ work? Is there an example?
A Lie $H$-pseudo algebra is a Lie algebra in the pseudotensor category $\mathcal{M}^*(H)$, but how does $S_I$ act on the composition of a pseudotensor category $\mathcal{M}^*(H)$?
The definition of pseudotensor category given by the book by A. Beilinson and V. Drinfeld, "Chiral algebras," didn't mention the equivariance axiom. Does the equivariance follow from the other properties they give? How?