Related question (I hoped to turn this into an answer for the user, but in the end it became a question on its own right!): this
In the book
Neeman, Amnon. Triangulated categories. No. 148. Princeton University Press, 2001.
there is a definition I would like to translate in the language of Lurie's stable $\infty$-categories: let me recall it in a couple of words.
Let $\cal T$ be a triangulated category admitting countable products and coproducts. We define the homotopy colimit of a diagram $X\colon\mathbb \omega\to \cal T$ (which consists of a sequence $X_0\xrightarrow{j_0} X_1\xrightarrow{j_1} X_2\to\dots$) as the object which completes the triangle $$ \coprod_{n\in\omega} X_n\xrightarrow{1-s}\coprod_{n\in\omega} X_n \to \underrightarrow{\text{holim}}_n X_n \to^+ $$ One can now prove some results: e.g. this construction is unique albeit non canonically. Everything can be dualized: the homotopy limit of the same sequence is the object $P[-1]$ obtained putting the arrow $$ \prod_{n\in\omega} X_n\xrightarrow{}\prod_{n\in\omega} X_n $$ into a distinguished triangle $\prod_{n\in\omega} X_n\xrightarrow{}\prod_{n\in\omega} X_n\to P\to^+$.
I feel uncomfortable with this construction as I'm not able to figure out what it should abstract. Has it something to do with a "telescope" construction?
And... what should a sensible translation of this construction be, in the setting of a stable $\infty$-category? Every piece of the construction makes sense (distinguished triangle $\to$ fiber-cofiber sequence, etc.), and yet it seems to have nothing to do with a "genuine" homotopy (co)limit.
