In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the **unseparated derived category** $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a Grothendieck abelian category. The unseparated derived category $\check{{\cal D}}({\cal A})$ is a stable presentable $\infty$-category equipped with a natural t-structure which is compatible with filtered colimits and whose heart is identified with ${\cal A}$. It is related to the usual derived $\infty$-category ${\cal D}({\cal A})$ by a t-exact functor $\check{{\cal D}}({\cal A}) \to {\cal D}({\cal A})$ which exhibits ${\cal D}({\cal A})$ as the "left separation" of $\check{{\cal D}}({\cal A})$. In addition, $\check{{\cal D}}({\cal A})$ itself enjoys the following universal property (see Theorem C.5.8.8 and Corollary C.5.8.9 of loc.cit): if ${\cal C}$ is any other stable presentable $\infty$-category equipped with a $t$-structure which is compatible with filtered colimits, then restriction along the inclusion of the heart $A \to \check{{\cal D}}({\cal A})$ induces an equivalence
$$ {\rm LFun^{{\rm t-ex}}}(\check{{\cal D}}({\cal A}),{\cal C}) \stackrel{\simeq}{\to} {\rm LFun^{{\rm ex}}}({\cal A},{\cal C}^{\heartsuit}) $$
where on the left hand side we have colimit preserving t-exact functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ and on the right hand side we have colimit preserving exact functors ${\cal A} \to {\cal C}^{\heartsuit}$. This is indeed a very satisfying universal characterization of $\check{{\cal D}}({\cal A})$ together with its t-structure. However, for various reasons it can be useful to have a universal characterization of $\check{{\cal D}}({\cal A})$ *without* the t-structure. To see how this might go, note that t-exact colimit preserving functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ send short exact sequences in ${\cal A}$ to cofiber sequences in ${\cal C}$ and filtered colimits in ${\cal A}$ to filtered colimits in ${\cal C}$. One may hence consider the possibility that restriction along ${\cal A} \to \check{{\cal D}}({\cal A})$ induces an equivalences between *all* colimit preserving functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ on the one hand and and all functors ${\cal A} \to {\cal C}$ which preserve filtered colimits and send short exact sequences in ${\cal A}$ to cofiber sequences in ${\cal C}$ on the other.

Question 1: Is this true?

A positive answer to Question 1 would imply that $\check{{\cal D}}({\cal A})$ admits the following explicit description: we would be able to identify it with the $\infty$-category of presheaves of spectra ${\cal A}^{{\rm op}} \to {\rm Sp}$ which send filtered colimits in ${\cal A}$ to filtered limits and send short exact sequences in ${\cal A}$ to fiber sequences of spectra.

As a variant to Question 1, one may hope that the connective part $\check{{\cal D}}({\cal A})_{\geq 0}$ enjoys the same universal characterization when ${\cal C}$ is now replaced with a Grothendieck prestable $\infty$-category, or maybe even any presentable $\infty$-category. Such a characterization would imply that $\check{{\cal D}}({\cal A})_{\geq 0}$ can be identified with the $\infty$-category of presheaces of **spaces** ${\cal A}^{op} \to {\cal S}$ which send filtered colimits to cofiltered limits and short exact sequences to fiber sequences of spaces.

Question 2: Is this true?

**Remarks:**
1) A positive answer to Question 2 would imply a positive answer to Question 1 since $\check{\cal D}({\cal A}) \simeq {\rm Sp}(\check{\cal D}({\cal A})_{\geq 0})$ is the stabilization of $\check{\cal D}({\cal A})_{\geq 0}$.

2) If ${\cal A} = {\rm Ind}(A_0)$ with $A_0$ an abelian category with enough projectives in which every object has finite projective dimension then $\check{\cal D}({\cal A}) \simeq {\cal D}({\cal A})$ (Proposition C.5.8.12 in loc.cit) and can also be described using complexes of projective objects. In this case $\check{\cal D}({\cal A})_{\geq 0} \simeq {\cal P}_{\Sigma}((A_0)_{{\rm proj}})$ is the $\infty$-category obtained from $(A_0)_{{\rm proj}}$ by freely adding sifted colimits. In particular, $\check{\cal D}({\cal A})_{\geq 0}$ admits a universal characterization as a presentable $\infty$-category and $\check{\cal D}({\cal A}) \simeq {\rm Sp}(\check{\cal D}({\cal A})_{\geq 0})$ admits a universal characterizations as a stable presentable $\infty$-category (without the t-structure). However, even in this case, I don't know how to deduce from this particular universal characterization the other universal characterization I'm interested in (the missing part is to show that a coproduct preserving functors $(A_0)_{{\rm proj}} \to {\cal C}$ extends in an essentially unique way to a functor ${\cal A}_0 \to {\cal C}$ which sends short exact sequence to cofiber sequences).