Alternate formulation of the question (I think): What's a precise version of the statement: "In a stable $\infty$-category, finite limits and finite colimits coincide"?

Recall that a stable $\infty$-category is a type of finitely complete and cocomplete $\infty$-category characterized by certain exactness conditions. Namely,

There is a zero object $0$, i.e. an object which is both initial and terminal.

Every pushout square is a pullback square and vice versa.

Item (2) takes advantage of a peculiar symmetry of the "square" category $S = \downarrow^\to_\to \downarrow$; namely $S$ can either be regarded as $S' \ast \mathrm{pt}$ where $S' = \cdot \leftarrow \cdot \rightarrow \cdot$ is the universal pushout diagram, or $S$ can be regarded as $S = \mathrm{pt} \ast S''$ where $S'' = \cdot \rightarrow \cdot \leftarrow \cdot$ is the universal pullback diagram. Hence it makes sense to ask, for a given $S$-diagram, whether it is a pullback, a pushout, or both. Item (1) similarly takes advantages of the identities $\mathrm{pt} = \emptyset \ast \mathrm{pt} = \mathrm{pt} \ast \emptyset$.

But I can't shake the feeling that notion of a stable infinity category somehow "transcends" this funny fact about the geometry of points and squares. For one thing, one can use a different "combinatorial basis" to characterize the exactness properties of a stable $\infty$-category, namely:

1.' The category is (pre)additive (i.e. finite products and coproducts coincide)

2.' The loops / suspension adjunction is an equivalence.

True, (2') may be regarded as a special case of (2) -- but it may also be regarded as a statement about the (co)tensoring of the category in finite spaces.

Both of these ways of defining stability say that *certain* limits and colimits "coincide", and my sense is that in a stable $\infty$-category, *all* finite limits and colimits coincide -- *insofar as this makes sense*.

**Question:**

Is there a general notion of "a limit and colimit coinciding" which includes

zero objects

biproducts (= products which are also coproducts)

squares which are both pullbacks and pushouts

suspensions which are also deloopings

and if so, is it true that in in a stable $\infty$-category, finite limits and finite colimits coincide whenever this makes sense?

I would regard this as investigating a different sort of exactness to the exactness properties enjoyed by ($\infty$)-toposes. In the topos case, I think there are some good answers. For one, in a topos $C$, the functor $C^\mathrm{op} \to \mathsf{Cat}$, $X \mapsto C/X$ preserves limits. Foir another, a Grothendieck topos $C$ is what Street calls "lex total": there is a left exact left adjoint to the Yoneda embedding. It would be nice to have similar statements here which in some sense formulate a "maximal" list of exactness properties enjoyed by (presentable, perhaps) stable $\infty$-categories, rather than the "minimal" lists found in (1,2) and (1',2') above.