I read Algebraic Operads by Loday and Vallette, namely paragraph 5.3.7 Partial definition of an operad. It's not clear to me how does $\sigma''$ act (the phrase "acting identically on ... with values in ..." sounds a bit wierd to me). Could you please write down an explicit formula for $\sigma''$ so that it would make sense?
Consider the simple example $\sigma=(123)(45)$, $m=6$, $n=4$ and $i=3$. The text suggests to me that $\sigma’’$ acts as follows:
We have $\{1,2,6,7,8,9\}\to\{4,5,6,7,8,9\}$, and $\sigma’’$ acts like $\sigma$, so sends the first element on the left to the second element on the right, the second element on the left to the third element on the right, and so on: $$ 1\mapsto5, \ 2\mapsto6, \ 6\mapsto4, \ 7\mapsto8, \ 8\mapsto 7, \ 9\mapsto9. $$ Also $\{3,4,5\}\to\{1,2,3\}$, and $\sigma’’$ acts like the identity, so sends the first element on the left to the first element on the right, and so on: $$ 3\mapsto1, \ 4\mapsto2, \ 5\mapsto3. $$ Putting this together we have $$ \sigma’’ = (153)(264)(78). $$