limits and products stable $\infty$-category In an abelian category $\mathcal{A}$, for a system $\{F_i,\phi_{ij}\}$ we have an exact sequence
$0\to \lim F_i\to \prod F_i \to \prod F_i$
where the second map is given by $id-\prod\phi_{ij}$. Is there a version of this for stable $\infty$-categories? Meaning, if $\mathcal{C}$ is a stable $\infty$-category and $F_i$ is a system in $\mathcal{C}$, then is there a fiber sequence in $\mathcal{C}$ given by
$\lim F_i\to \prod F_i \to \prod F_i$?
In case $\mathcal{C}=\mathcal{D}^+(A)$ where $A$ is an abelian category with exact products, enough invectives and the system is countable, then this is true, see for example Stacks Project Lemma 0BK7. I was wondering if one can conclude the same for a more general $\mathcal{C}$
 A: In the case of an $\mathbb N^{op}$-indexed system specifically, the answer is yes (note that this is implicit in the Stacks project link you gave); in fact if you replace "fiber sequence" by "equalizer", this holds in an arbitrary $\infty$-category with the appropriate limits (namely products and equalizers). The description in terms of fibers does not hold in general though (in an arbitrary $\infty$-category with limits, the "fiber" without specifying a basepoint does not even make sense)
There are several ways a proof could go. One approach is to use the fact that $\mathbb N$, as an $\infty$-category, is the infinite pushout $[1]\coprod_{[0]}[1]\coprod_{[0]}[1] \dots$. Another approach is to use the Yoneda lemma to reduce to the case of the $\infty$-category of spaces, and there, use explicit models for homotopy limits (this is probably simpler to actually write down, if not as conceptual).
For a general filtered poset $I$, however, homotopy limits over $I^{op}$ can be more complicated.
There is always the general "Bousfield-Kan formula", which expresses $\lim_{I^{op}}$ in the form of the totalization of a cosimplicial object (the $\infty$-analogue of an equalizer): namely, $\lim_{I^{op}}F$ can be described as the limit over $\Delta$ of a functor that looks like $[n]\mapsto \prod_{i_0\to ... \to i_n} F(i_0)$ .
