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.