  From a map of operads $\eta : O\rightarrow O'$, we introduce a cofibrant
replacement of the operad $O$ in the category of bimodules over itself such
that the corresponding model of the derived mapping space of bimodules
$Bimod_{O}^{h}(O;O')$ is an algebra over the one dimensional little cubes
operad $\mathcal{C}_{1}$. In the present work, we also build an explicit weak
equivalence of $\mathcal{C}_{1}$-algebras from the loop space $\Omega
Operad^{h}(O;O')$ to $Bimod_{O}^{h}(O;O')$.
