I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by $$ \small (f\bullet g)(a_1,\dots,a_{p+q-1})=\sum\limits_{i=0}^{p} (-1)^{i(q+1)} f(a_1,\dots,a_i,g(a_{i+1},\dots,a_{i+q}),a_{i+q+1},\dots,a_{p+q-1}) $$ where $f$ is a $p$-cochain and $g$ is a $q$-cochain. After that he defines the Gerstenhaber bracket by $[f,g]_G=f\bullet g-(-1)^{(p-1)(q-1)}g\bullet f$.

I have noticed that the summands in the operation $\bullet$ look a lot like the partial composition in operads, as if $f$ was an element of arity $p$ and $g$ was an element of arity $q$ in some operad. So my question is, **is it just a coincidence, or there is indeed some way to see $f,q$ as operations in some operad?** Or maybe **is there a way to see the operation $\bullet$ coming from some operadic context?**

I am a complete novice in these things, so I am sorry if my question is silly. Anyways I would be happy if someone would explain that to me.

Thank you for your help!