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 categoryis 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, satisfyingAssociativity,Unit, andEquivarianceaxioms.

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?