The original definition of operads involves maps
$$ \gamma\colon C(n) \times \prod\_{i=1}^n C(k\_i) \to C(\sum\_ik\_i) $$ 
There is an alternative definition in terms of maps
$$ \circ\_i \colon C(n) \times C(m) \to C(n+m-1). $$
It is not hard to outline an argument that the two definitions are equivalent (subject to some assumptions about $C(0)$ and $C(1)$; various choices of details are possible). Have the details been spelled out carefully somewhere? I have seen this called an "observation", but that seems a little blasé to me.



Detailed proof that $\circ_i$ operads are the same as operads
There is a detailed discussion of the "partial" composition operations of an operad in Volume 1, Chapter 2 of Fresse's monograph:


* Benoit Fresse, *Homotopy of Operads & Grothendieck-Teichmüller Groups*. ([book project web page](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/)) ([author pdf for volume 1](http://math.univ-lille1.fr/~fresse/OperadHomotopyBook/OperadHomotopy-FirstVolume.pdf))


The equivalence of the two definitions of operad is stated as Theorem 2.1.10, and the proof is worked out in Appendix A (see A.2.10).


