Right action by an operad on a non symmetric collection

Question

Suppose we have a non symmetric operad $\mathcal{O}$, a collection of sets $\{P(n)\}_{n\geq 0}$ and maps

$$P(n)\otimes \mathcal{O}(k_1)\otimes\cdots \otimes \mathcal{O}(k_n)\to P(k_1+\cdots + k_n)$$

satisfying some conditions on associativity when composed with the operadic composition, etc.

Question. Is there a name for the collection $P$ equipped with this structure?

It is not an algebra over the operad, nor does it seem to fit into any definition of a module I have come across. We assume no further structure on $P$.

Answer 1

This structure is called a right module over the operad. You can have a look at the book "Modules over operads and functors" by Benoit Fresse for a thorough study of these objects.