When does the loop functor $\Omega^\infty:Sp(C) \rightarrow C$ commute with filtered colimits?

Let $$C$$ be a pointed $$\infty$$-category which admits finite limits.

Let $$Sp(C)$$ denote the $$\infty$$ category of spectrum objects. One way to define, i.e. 1.4.2.24, is by taking the homotopy limit in $$Cat_\infty$$, the $$\infty$$-category of categories. $$Sp(C):= \varprojlim \left( \cdots \xrightarrow {\Omega} C \xrightarrow {\Omega} C \right)$$

Let us denote $$\Omega^\infty: Sp(C) \rightarrow C$$ as the projection onto the last component.

I would like to understand what categorical properties of $$\Omega^\infty$$ satisfy. My question is

If each $$\Omega$$ commute with $$I$$-indexed limit does this imply $$\Omega^\infty$$ does too?

The reason I am concerned with this question: It is claimed in C.1.4.1, that

if $$C$$ is a prestable and presentable $$\infty$$-category and $$\Omega:C \rightarrow C$$ commutes with filtered colimits then $$\Omega^\infty$$ commutes with filtered colimits.

A prestable $$\infty$$-category by definition can be intriscally characterized, C.1.2.1 as a category which satisfies the following conditions

• pointed and admits finite colimits.
• suspension is fully fiathful
• every morphism $$Y \rightarrow \Sigma Z$$ lies in a pullback square with top right part $$X \rightarrow Y \rightarrow \Sigma Z$$ and bottom left $$0$$. Further, the sequence $$X \rightarrow Y \rightarrow \Sigma Z$$ is a cofiber sequence.

I have recorded my thoughts below, which one may safely ignore.

Both strategies I know do not really apply - these are based on the case $$C=S_*$$, the $$\infty$$-cat of pointed spaces.

Strategy 1. $$\Omega^\infty: Sp(S) \rightarrow S_*$$. $$\Omega^\infty$$ is corepresented by $$\mathbb{S}=\Sigma^\infty S^0$$, the sphere spectrum, where we $$\Sigma^\infty$$ is left adjoint of $$\Omega^\infty$$. Now by noting that that $$S^0$$ is a compact object in $$S_*$$ result follows.

Strategy 2. Consider the $$\infty$$-cat $$Pr^\omega$$ of compactly generated, in the sense of 5.5.7.1, $$\infty$$-categories with right adjoints. We prove that $$S\in Pr^w$$ and that $$Pr^w \hookrightarrow Cat_\infty$$ reflects (filtered) limits.

The result is true, more generally, if you take a class of diagrams $$\mathcal K$$ and the $$\infty$$-category $$\widehat{Cat_\infty}^\mathcal K$$ of $$\infty$$-categories that have all $$\mathcal K$$-indexed colimits, and functors between them that preserve those, then the forgetful functor $$\widehat{Cat_\infty}^\mathcal K\to \widehat{Cat_\infty}$$ preserves all limits, in fact it has a left adjoint.

This is stated as Corollary 5.3.6.10. in Lurie's Higher Topos Theory (with his notations, $$\mathcal K' =$$ my $$\mathcal K$$, and his $$\mathcal K= \emptyset$$).

From this, your result follows, as if $$\Omega$$ preserves $$I$$-indexed colimits, then your diagram lives in $$\widehat{Cat_\infty}^{\{I\}}$$, so its limit does too, and the projection functors too, in particular $$\Omega^\infty: Sp(C)\to C$$ is one of those projection functors, so it preserves $$I$$-indexed colimits (this is, of course, assuming that $$C$$ has all $$I$$-indexed colimits - which is the case in the statement you refer to, as of course a presentable $$\infty$$-category has all filtered colimits)

Your strategy 1 is in this sense somehow misguided, as proving that $$\mathbb S$$ is compact essentially uses that $$\Omega^\infty$$ preserves filtered colimits.

Actually, a less general, but perhaps easier proof works in the special case of $$Sp(C)$$ and filtered colimits: $$Sp(C)$$ can be seen as a certain full subcategory of $$Fun(\mathbb{Z\times Z},C)$$ (such a functor is a grid, $$Sp(C)$$ is the full subcategory on those grids that are only $$0$$ objects off the diagonal, and such that certain squares are pullbacks), and $$\Omega^\infty$$ is then simply the restriction to this subcategory of the evaluation at $$0$$.

Now if $$C$$ has all filtered colimits, and $$\Omega$$ commutes with those, then $$Sp(C)\subset Fun(\mathbb{Z\times Z},C)$$ is closed under filtered colimits (the only pullbacks appearing in its definition are pullbacks defining $$\Omega$$), so that, as in functor categories in general, filtered colimits in $$Sp(C)$$ are computed pointwise; and so in particular $$\Omega^\infty$$ commutes with those.

This second proof is less general, but it's easier and gets you what you want- and perhaps it allows for a better understanding in this specific context ?