In Berger-Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (1.5.6) there is a description of the operad, whose algebras are non-colored symmetric operads.
I have some trouble understanding how the composition law in this operad works.
Is it true that in the result of composition $(T,\sigma, \tau)$ of some labeled trees $(T_i,\sigma_i,\tau_i)$ the planar tree $T$ and the labeling of vertices $\sigma$ depend only on $T_i$ and $\sigma_i$?
What is the correct labeling of input edges of trees on the right side?