I am studying Algebraic Operads, following the book Algebraic Operads by Jean-Louis Loday and Bruno Vallette.
In this book they provide many equivalents definitions for algebraic operads, and
I am particularly interested about the monoidal and classical definitions (May's definition).
In the Proposition 5.3.2 we get the equivalence of the classical and monoidal definitions.
Now my focus is on the item (a) :
I don't know what the two underlined terms means. But, reading May's definitions I suppose that
the item (a) refers to:
Where $\gamma$ are the linears aplications that define operads struture in the classical sense.
Is that correct? If it is, what does mean $\sigma \otimes \sigma^{-1}$? I think $\sigma \otimes \sigma^{-1}$ leave $x_n\otimes x_{k_1} \otimes \cdots x_{k_n} $ in $\sigma\cdot x_n\otimes x_{\sigma^{-1}(k_1)}\otimes \cdots \otimes x_{\sigma^{-1}(k_n)} $. Where $\cdot$ denote the action of $\mathbb{S}_n$ (or $\sum_{n}$) in $\mathcal{P}(n)$.