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 $\mathcal{C}$