I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions?
The typical example is the following. Let $R$ be a connective $\mathbb E_1$-ring. Then we say that a connective right $R$-module $M$ is flat [Lurie's Higher Algebra, Def 7.2.2.10] if the right $\pi_0(R)$-module $\pi_0(M)$ is flat, and the canonical map $\pi_0(M)\otimes_{\pi_0(R)}\pi_n(R)\to\pi_n(M)$ is an isomorphism for every $n\in\mathbb Z$.
Lazard's theorem tells us that a connective right $R$-module $M$ is flat if and only if it can be written as a filtered colimit of finite free right $R$-modules. This generalizes to prestable $\infty$-categories generated by compact projective objects:
Definition 1. Let $\mathcal M_{\ge0}$ be a projectively generated prestable $\infty$-category. We say that an object $M\in\mathcal M_{\ge0}$ is Lazard-flat if it can be written as a filtered colimit of compact projective objects of $\mathcal M_{\ge0}$.
Another characterization is in terms of tensor products: a connective right $R$-module $M$ is flat if and only if the tensor product $\operatorname{LMod}_R\to\operatorname{Sp},N\mapsto M\otimes_RN$ carries the heart into the heart.
This should also generalize to certain prestable $\infty$-categories. The key observation is that $\operatorname{LMod}_R$ is the dual of $\operatorname{RMod}_R$, which leads to the following attempt:
Definition 2. Let $\mathcal M_{\ge0}$ be a dualizable Grothendieck prestable $\infty$-category. Let $\mathcal M^\heartsuit$ denote its heart. We say that an object $M\in\mathcal M_{\ge0}$ is flat if the composite functor $\mathcal M^\vee\xrightarrow{(-)\boxtimes M}\mathcal M^\vee\otimes\mathcal M\xrightarrow{\operatorname{ev}}\operatorname{Sp}$ carries $(\mathcal M^\heartsuit)^\vee$ to $\operatorname{Ab}=\operatorname{Sp}^\heartsuit$.
I wonder whether these concepts are studied and compared in the literature, or whether there are other flatness which could be compared with these two?