The universal property of the unseparated derived category 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).  
 A: Yes, both of these statements are true (I thought they were in the book, but I can't seem to find them now).
Here is a proof sketch. Let's start with the case described in 2).
Let $\mathcal{C}$ be any presentable $\infty$-category. As noted in the question, what you need to identify are functors $F: \mathcal{A}_0 \rightarrow \mathcal{C}$ that preserve finite coproducts and
carry short exact sequences to cofiber sequences to functors $F_0: \mathcal{A}^{proj}_0 \rightarrow \mathcal{C}$ that preserve finite coproducts. There's an obvious restriction functor in one direction, and there's a functor of left Kan extension in the other (the universal property of $\mathcal{D}( \mathcal{A} )_{\geq 0}$ guarantees that this left Kan extension yields a functor that behaves well on exact sequences). The thing you need to check is that any $F: \mathcal{A}_0 \rightarrow \mathcal{C}$ as above agrees with the left
Kan extension of its restriction to $\mathcal{A}_0^{proj}$; let's denote that functor by $F'$, so there's a natural transformation $F' \rightarrow F$ which is an equivalence on projective objects. Since both $F$ and $F'$ carry short exact sequences to cofiber sequences, the collection of objects $X$ for which $F'(X) \rightarrow F(X)$ is an equivalence is closed under taking cokernels of injective maps, and therefore (by induction) contains all objects of finite projective dimension, which is all objects of $\mathcal{A}_0$ by assumption.
To handle the general case, we can use the fact that every Grothendieck abelian category can be obtained as a quotient $\mathcal{A} / \mathcal{B}$, where $\mathcal{A}$ is as above and $\mathcal{B}$ is some localizing subcategory. Let $\mathcal{D}_0$ be the smallest localizing subcategory of $\check{\mathcal{D}}( \mathcal{A})_{\geq 0}$ which contains $\mathcal{B}$, so that we can identify $\check{\mathcal{D}}( \mathcal{A} / \mathcal{B} )_{\geq 0}$ with the quotient $\check{\mathcal{D}}( \mathcal{A} )_{\geq 0} / \mathcal{D}_0$. The first part of the proof shows that the following data are equivalent:
(1) Functors $F: \mathcal{A} \rightarrow \mathcal{C}$ which preserve finite coproducts, carry short exact sequences to cofiber sequences, and preserve filtered colimits.
(2) Functors $F^{+}: \check{\mathcal{D}}(\mathcal{A})_{\geq 0} \rightarrow \mathcal{C}$ which preserve small colimits.
We wish to show that if $F$ and $F^+$ correspond under this equivalence, then the following conditions are equivalent:
(a) For every map $u: X \rightarrow Y$ in the abelian category $\mathcal{A}$, if the kernel and cokernel of $u$ belong to $\mathcal{B}$, then $F(u)$ is an equivalence (so that $F$ factors through $\mathcal{A} / \mathcal{B}$).
(b) For every map $u: X \rightarrow Y$ in $\check{\mathcal{D}}(\mathcal{A})_{\geq 0}$, if the cofiber belongs to $\mathcal{D}_0$, then $F^{+}(u)$ is an equivalence in $\mathcal{C}$ (so that $F^{+}$ factors through
$\check{\mathcal{D}}( \mathcal{A} )_{\geq 0} / \mathcal{D}_0$).
The implication $(b) \Rightarrow (a)$ is immediate. To proceed in the reverse direction, let $u: X \rightarrow Y$ be as in $(b)$ and choose an object $Y'$ in $\mathcal{A}$ and a map $Y' \rightarrow Y$ which is surjective on $\pi_0$, and set $X' = Y' \times_{Y} X$. Then $u$ is a pushout of the projection map $u': X' \rightarrow Y'$, so it suffices to show that $F^{+}(u')$ is an equivalence. The map $u'$ factors as a composition $X' \xrightarrow{w} Y'_0 \xrightarrow{v} Y'$ where $v$ is a monomorphism in $\mathcal{A}$ and $w$ is surjective on $\pi_0$. Using the assumption that the cofiber of $u$ belongs to $\mathcal{D}_0$, it is not hard to check that $Y' / Y'_0 \in \mathcal{B}$ and that the fiber of $w$ belongs to $\mathcal{D_0}$. Then $F^{+}(v)$ is an equivalence by virtue of $(a)$, and $w$ is a pushout of the map $fib(w) \rightarrow 0$. 
We are therefore reduced to proving that the functor $F^{+}$ annihilates every object of $\mathcal{D}_0$, which is not hard to check (using the fact that it annihilates every object of $\mathcal{B}$).
