I'd like to know what's the name (if any) of the following categorical structure, and also references where it has been considered.

Given a category $C$, let $O=\{O(n)\}_{n\geq 0}$ be a sequence of functors $O(n)\colon C\times\stackrel{n}\cdots\times C\rightarrow C$ equipped with natural transformations $O(n)(O(p_1),\dots,O(p_n))\Rightarrow O(p_1+\cdots + p_n)$ satisfying the usual relations (as for operads). (A unit natural transformation $id_C\Rightarrow O(1)$ may be added.)

If $C$ is symmetric monoidal and $P$ is an operad, then the sequence of functors $(X_1,\dots,X_n)\mapsto P(n)\otimes X_1\otimes\cdots\otimes X_n$ fits in the previous setting.

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