In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism $$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$ where $X$ and $Y$ are geometric stacks and $\mathbf{QC}(X)$, $\mathbf{QC}(Y)$ are their categories of quasicoherent sheaves. Moreover, if $X$ is noetherian then we have an equivalence $\mathbf{QC}(X) \cong \mathbf{Ind}(\mathbf{Coh}(X))$.

Now suppose $X$ is a DG indscheme in Gaitsgory's sense, that is an ind-object $\varinjlim_{\alpha}X_{\alpha}$ in the category of $DG$ schemes such that all the transition maps $X_{\alpha} \to X_{\beta}$ are closed embeddings. My question is as follows. For nice enough indschemes $X$ and $Y$ (e.g. quasicompact and noetherian components), do we have a natural equivalence $$Map(X,Y) \cong Map(\textbf{IndCoh}(Y), \textbf{IndCoh}(X))?$$ I believe this should follow from Lurie's result as long as the $X_{\alpha}$'s and $Y_{\beta}$'s satisfy Tannaka duality, since in that case we have $$\mathbf{IndCoh}(X) \cong \varinjlim_{\alpha}\mathbf{IndCoh}(X_{\alpha}) \cong \varinjlim_{\alpha}\mathbf{QC}(X_{\alpha})$$


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