I am now reading the book ``Algebraic Operads'' written by Murray R. Bremner and Vladimir Dotsenko. In this book, on page 114-115, Definition Let $S_n$ be a permutation group. A twisted associative algebra is a nonnegatively graded associative algebra $A=\oplus_{n\geq0}A(n)$ for each graded component $A(n)$ is a right $S_n$-module, and each product map
$$\mu_{n_1, n_2}:A(n_1) \otimes A(n_2)\rightarrow A(n_1+n_2)$$
is a morphism of $S_{n_1} \times S_{n_2}$-modules.
The author said the map $$A(n_1) \otimes A(n_2)\rightarrow A(n_1+n_2)$$
is a morphism of $S_{n_1} \times S_{n_2}$-modules is equivalent to saying that
where $\text{sh }(n_1, n_2)$ is the $(n_1, n_2)$-shuffle, $p$ is the canonical projection and $f$ is a symmetric group action.
I don't understand this equivalent completely and who can help me ? I also don't know why my tex code is not showed here correctly.