@user40276 no it's false: that colimit needs to be taken in something like the category of presentable categories in order to compute the right answer, so it doesn't even make sense unless C is presentable.
the colimit in (infty-)categories produces the spanier whitehead category, which has the wrong mapping spaces.
@DylanWilson Thanks for the reply. That's weird somehow. I thought that the unit and counit would induce an invertible homotopy between the squares in the (co)limit (co)cones. Still, I should have know that it shouldn't be the case since the colimit will never have non-connective stuff.
@user40276 I think you mean that it cannot contain stuff that's not bounded below :). The SW category is stable, so contains all desuspensions of connective stuff
Is there someone here who read the proof of Browder's theorem and is willing to help me understand the structure of the proof? From what I understand so far the game seems to be to detect the kervaire invariant 1 framed manifolds in "wu-bordism" then show they all have to be on the 2 line of the ASS then compute the 2-line of of the ASS for wu-bordism. Is that true? Even if this is true I have a very hard time identifying the arguments in Browder's paper which prove each step...
@user40276 Related question: https://mathoverflow.net/questions/210838/does-the-forgetful-functor-from-presentable-infty-categories-to-infty-cate/210852#210852
(That this fails for $\mathrm{Sp}$ is given as a counterexample in the answer)
@SaalHardali i don't know, but i'm willing to make a guess. the adams spectral sequences based on HF2 and on MO look the same, so i imagine "M in Omega^fr(*) has image in MO-Adams filtration n" might mean something along the lines of "there is a sequence of maps of framed manifolds M --> M1 --> ... --> Mn --> * where each Mj --> M(j+1) is nontrivial as an element of Omega^O_*(M(j+1)). that's probably not right, but i also wouldn't expect it to be far off
