Tannaka duality for $DG$ indschemes

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 $$\mathbf{IndCoh}(X) \cong \varinjlim_{\alpha}\mathbf{IndCoh}(X_{\alpha}) \cong \varinjlim_{\alpha}\mathbf{QC}(X_{\alpha})$$