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. 



What is this operad-like structure called?
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](http://arxiv.org/pdf/math/0305049v1.pdf) 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.


