I'm asking here this question I asked on MSE that got no answers.

Let $V$ be a dg-module and $P$ an operad. The free $P$-algebra on $V$ is defined by $P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\otimes r})_{\Sigma_r}$, where the $Σ_r$-quotient identifies tensor permutations with the action of permutations on $P(r)$.

On the other hand, $V$ is said to be a $P$-algebra if there is a morphism of operads $P\to End_V$, where $End_V$ is the endomorphism operad of $V$. Equivalently, $V$ is a $P$-algebra if there is a collection of maps $P(r)\otimes V^{\otimes r}\to V$ satisfying certain conditions.

How do these two notions reconcile?

An element of $p\otimes x_1\otimes\cdots \otimes x_r\in P(V)$ can be written as $p(x_1\otimes\cdots\otimes x_r)$ and therefore $p$ is interpreted as a map $V^{\otimes r}\to V$. But how can we realize $p$ as an element of $End_V(r)$ so that we do have the map of operads $P\to End_V$? Or equivalently, how can identify $p(x_1\otimes\cdots\otimes x_r)$ with an element of $V$ so that we have the maps $P(r)\otimes V^{\otimes r}\to V$?