The operadic butterfly is a diagram in the category of operads in vector spaces. It extends the short exact sequence relating commutative, associative and Lie operads.


$$\begin{array}{ccccc}
 & Dend & & & & Dias & \newline
 \nearrow & &\searrow & &\nearrow & &\searrow\newline
Zinb & & &Ass & & & \quad Leib \newline
 \searrow & &\nearrow & &\searrow & &\nearrow\newline
 & Comm & & & & Lie & \newline 
\end{array}$$


Here is a [paper](http://arxiv.org/abs/math/0409183) by Loday in which some discussion can be found. As the reference explains, the Koszul duality functor, $\mathcal{O}\mapsto \mathcal{O}^!$, gives the above diagram symmetry about the vertical axis.


Is the butterfly also symmetric about the horizontal axis? 


In other words, does there exist a functor $F : Operad(C) \to Operad(C)$, where $C$ is vector spaces or chain complexes, such that $F\circ F \simeq 1$, $F$ fixes the operads: $Zinb$, $Ass$ and $Leib$, and exchanges the pairs: $(Dend,Dias) \leftrightarrow (Comm,Lie)$.



Is the operadic butterfly symmetric?
I have never seen such a functor and I doubt that it would exist. For instance, the operad Com is symmetric and one-dimensional in every arity. Whereas the operad Dend is regular (coming from a non-symmetric one) and is generated by two (non-symmetric) generators. I also now no non-trivial functor which preserves Zinb or Leib. 


Can I ask why you are looking for such a functor (despite the symmetric shape of the operadic butterfly)? 


