TITLE: Detailed proof that $\circ_i$ operads are the same as operads
QUESTION [9 upvotes]: 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.

REPLY [7 votes]: 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) (author pdf for volume 1)

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).