@YonatanHarpaz Of course, my last question makes sense only if you do not start with an equivalence already, but with just an exact endofunctor $F$ of $\mathbf{Sp}$ with an equivalence $F(\mathbb{S})\rightarrow \mathbb{S}$. So the properly stated follow-up question was: is it true that, in the infinity category setting, an exact (colimit preserving?) endofunctor $F$ of $\mathbf{Sp}$ equipped with an equivalence $F(\mathbb{S})\rightarrow \mathbb{S}$ is in fact equivalent to the identity of $\mathbf{Sp}$?

@TomGoodwillie Good point, I do not know what I was thinking. I will edit the question in a second.

@YonatanHarpaz Thanks for your answer! I'd still be interested in an answer about the homotopy category, but the infinity category approach is still interesting. Do you have references to suggest? Also, wouldn't your functor $F$ need to be colimit-preserving rather than just exact for the result about extending an equivalence $F(\mathbb{S})\rightarrow \mathbb{S}$ to hold?

[...] then $\text{hcolim}FQX\rightarrow \text colim FQX$ is a weak equivalence.

@ZhenLin This probably sounds silly to you, but I can not exactly understand how that fact should imply preservation of homotopy colimits as in (1) (and this is precisely the source of my questions). I know that, when $F$ is a left Quillen functor, $FQ\text{hcolim}$ and $\text{hcolim} FQ$ are (canonically) naturally weakly equivalent as functors over $F\text{colim}\cong\text{colim} F$, for any cofibrant replacement functor Q, but I do not see how to deduce from this that, if $X\colon\mathscr{I}\rightarrow\mathscr{M}$ is a hcolim diagram,[...]

@ZhenLin I would still be interested in a complete answer (or in a reference to an answer) to questions (1) and (2) just for homotopy colimit or limit diagrams as well, i.e. for those diagrams $X\colon\mathscr{I}\rightarrow\mathscr{M}$ such that $\text{hcolim}X\rightarrow \text{colim}X$ or $\text{lim}X\rightarrow \text{hlim}X$ is a weak equivalence.

@TylerLawson Indeed, I am not.

[...] In giving that definition I was motivated by the usual notion of homotopy pushout and homotopy pullback squares, where, for the homotopy pullback case, say, one requires to a commutative square to be such that the canonical map from the top left corner to the homotopy limit of the cospan consisting of the remaining three vertices is a weak equivalence. To me, it seems like the only meaningful "canonical map" is the one factoring through the ordinary limit of that cospan so that the notion of homotopy pullback seemed to be a particular instance of my notion of homotopy limit cone...

Dear @ZhenLin, I guess I can see what you would like to suggest. However, I do not fully understand what you mean when you say that the definition is not "correct". Maybe I should have said it before, but I used the terminology "homotopy colimit cocone" just because I had to find a way to call a cocone having that property. However, I am not claiming that my definition should coincide with another one that perhaps is already conventional (I do not know if this is the case).[...]

@მამუკაჯიბლაძე Thank you very much for your attention and your efforts! However, I have edited my question trying to explain why I do not see how what you say should suffice (that would have been too long for a comment). I am not criticizing at all on what you say, of course, I am just trying to see if I can understand the point. And thanks again!

@მამუკაჯიბლაძე Nope, not even that approach seems to work. I thought I was able to get the needed morphisms but then I realized that I badly confused myself and interchanged $\text{Ho}(\mathbf{M}^{I})$ and $\text{Ho}(\mathbf{M})^I$ at some point. Just to mention one thing, I still can not see how to get the maps $\text{hocolim} Y\rightarrow\bar{Y}$, even in $\text{Ho}(\mathbf{M})$, without passing to the overlying model category, hence fixing some specific models for hocolim, thus destroying any hope of getting a canonical construction...