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 $$ \circ_i \colon C(n) \times C(m) \to C(n+m1). $$ 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.
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$\begingroup$ I've seen it in the Ginzburg & Kapranov Koszul duality paper. Not making this an answer since I am not sure what is your rigor standard :D (The paper has a subsequent erratum but it concerns a different place) $\endgroup$ – მამუკა ჯიბლაძე Sep 2 '16 at 9:33

1$\begingroup$ Phil Hackney just mentioned that they are not equivalent if $C(1)$ is not the unit of the monoidal structure. Maybe that is relevant? $\endgroup$ – Sean Tilson Sep 2 '16 at 9:57

$\begingroup$ @SeanTilson you are right that this is an issue; I have added a parenthetical comment. $\endgroup$ – Neil Strickland Sep 2 '16 at 10:17

11$\begingroup$ I put a proof in one of my papers, but the referee told me to take it out. $\endgroup$ – Gregory Arone Sep 2 '16 at 11:16

4$\begingroup$ @GregoryArone If you still have the draft of that paper you could copypaste that proof here as an answer and finally get it out in the open. (A reference to a MO post is better than nothing...) $\endgroup$ – Denis Nardin Sep 2 '16 at 12:31
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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 & GrothendieckTeichmüller 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).