TITLE: What is this operad-like structure called?
QUESTION [16 upvotes]: 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.

REPLY [11 votes]: If I'm getting right what you mean by "satisfying the usual relations (as for operads)" then your structure is what Tom Leinster calls in his Higher Operads, Higher Categories book an unbiased lax monoidal category. 
They can be seen as (lax) algebras for a $\mathbf{Cat}$-enriched operad:
First, start with the terminal object $1$ in $\mathbf{Set}^{\mathbb{N}}$ and build the free $\mathbf{Set}$-operad on it $F1$; this is generated by one operation per arity, but you get an awful lot of freely derived operations: $F1(n)$ is the set of planar trees with $n$ leaves. Now, make all the sets $F1(n)$ categories by taking the indiscrete category on them $I(F1(n))$. $I\colon \mathbf{Set} \to \mathbf{Cat}$ preserves products so this gives a $\mathbf{Cat}$-operad; lax algebras for it are unbiased lax monoidal categories. 
All this is in Leinster's book, in the chapter about algebraic definitions of monoidal categories.