Because I’m primarily interested in this question from the point of view of $ \infty$ -categories (in this case, modeled by quasicategories), I’ll ask this question using that terminology. In particular, I’ll just say “category” instead of $ \infty$ -category or quasicategory, to keep things legible. I suspect that, up to adding in the word “homotopy,” a lot of what I’m going to say holds very generally.

It’s not hard to work out that if $ Top$ is the category of $ \infty$ -groupoids (or anima, according to Peter Scholze), then there is an equivalence between comodules over any $ X\in Top$ , with respect to the Cartesian monoidal structure, and the slice category $ Top/X$ . I’ll write $ Comod_X(Top)\simeq Top/X$ . Furthermore, by Lurie’s straightening/unstraightening (a.k.a. the $ \infty$ -categorical Grothendieck construction), we have an equivalence $ Top/X\simeq Fun(X,Top)$ . The first category, $ Comod_X(Top)$ , is clearly comonadic over $ Top$ via the forgetful functor $ U\colon Comod_X(Top)→Top$ . Moreover, the colimit functor $ Fun(X,Top)→Top$ factors through $ Comod_X(Top)$ and is the left adjoint of an adjoint equivalence realizing the composite equivalence $ Comod_X(Top)≃Fun(X,Top)$ . Thus, $ Fun(X,Top)$ is comonadic over $ Top$ via the colimit functor $ colim\colon Fun(X,Top)→Top$ .

My question is whether or not this is more generally true, perhaps for some generic abstract reasons that I’m not aware or, or not seeing. Note that it is *not* in general true (I don’t think) that $ colim:Fun(D,C)→C$ is comonadic for arbitrary categories $ D$ and $ C$ . However, I’m happy with assuming that $ D$ is an $ ∞$ -groupoid and that $ C$ is “nice,” i.e. at least (locally) presentable.

So really the question is, are there check-able conditions under which the colimit functor $ colim:Fun(X,C)→C$ is comonadic, under the assumptions that $ X$ is an $ ∞$ -groupoid and $ C$ is presentable?