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

Delooping derived mapping spaces of bimodules over an operad