Derived mapping spaces between little $d$-disks operads $E\_d$ play an important role in embedding calculus. For example, [Dwyer-Hess](https://projecteuclid.org/euclid.gt/1513732411) expresses the homotopy of framed long knots as loop spaces such mapping spaces, a result which was generalized by [Boavida de Brito-Weiss](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/topo.12048).


When stating these results, the $E\_d$-operad is usually *unitary*: its space of 0-ary operations is contractible (in fact a point). There is also a *non-unitary* version $E\_d^{nu}$, where replace the $0$-ary operations by an empty set. There is a natural forgetful map from the derived mapping space of unitary operads to that of non-unitary operads


$$Map^h\_{\mathsf{Op}}(E\_m,E\_n) \to Map^h\_{\mathsf{Op}}(E\_m^{nu},E\_n^{nu}).$$



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> Is this a weak equivalence?
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This is known if we assume $m=n$ *and* we replace the operads by their rationalizations, by Section 7 of [Fresse-Willwacher](https://arxiv.org/abs/2003.02939).



Maps between unitary little disks operads and non-unitary little disks operads
A positive answer is the main theorem of my paper with Krannich and Horel, [Two remarks on spaces of maps between operads of little cubes](https://arxiv.org/abs/2211.00908). The proof uses a result of Haugseng and Kock to reduce it to a theorem of Lurie.


