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One of the remarkable results in Toen's paper is the existence of internal homs of dg-categories. Actually for two dg-categories $A$ and $B$, there exists a dg-category $RHom(A,B)$ such that for any dg-category $C$ we have $$ [C\otimes^LA,B]\cong [C,RHom(A,B)], $$ where $[-,-]$ means the map in Hqe, the homotopy category of dg-Cat after inverting all quasi-equivalences.

From the definition it is clear that for $B\simeq C$ in Hqe, we must have $$RHom(A,B)\simeq RHom(A,C)$$ in Hqe for any dg-category $A$.

My question is whether the inverse is true, i.e. for $A$, $B$, and $C$ dg-categories where $A$ is not quasi-equivalent to the empty category. Does $RHom(A,B)\simeq RHom(A,C)$ imply $B\simeq C$ in Hqe? If not, do we have counter examples?

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Actually the hom-set $[B,C]$ between two dg-categories in $\mathrm{Hqe}$ is described as \begin{equation} [B,C] \cong \mathrm{Iso}(H^0(RHom(B,C))),\end{equation} so if $RHom(A,B) \cong RHom(A,C)$ in $\mathrm{Hqe}$ for any $A$ and functorially in $A \in \mathrm{Hqe}$ (this is needed, I believe) then yes you can deduce that $B \cong C$ in $\mathrm{Hqe}$, by the ordinary Yoneda lemma.

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  • $\begingroup$ It is a good result. But I was asking if it is true for any one fixed nonzero dg-category $A$ and now I believe there are counter-examples. $\endgroup$
    – Zhaoting Wei
    Commented Dec 4, 2016 at 3:55

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