Let $\mathcal{C}$ be a presentable category, and let $S$ be a set of objects such that $S$ generates $\mathcal{C}$ under colimits, i.e., such that the smallest cocomplete subcategory of $\mathcal{C}$ containing $S$ is all of $\mathcal{C}$. Under what conditions is it true that for every object $x \in \mathcal{C}$, there exists a functor $f: \mathcal{I} \to \mathcal{C}$ whose image is contained in $S$ and whose colimit is isomorphic to $x$? This is true for example if $\mathcal{C}$ is a presheaf category and $S$ the canonical set of generators (the representables). I suspect the answer is no in general, but I don't know how to construct an example. 


I am also interested in the analogous question for presentable $\infty$-categories.


**Edit:** After posting this question, I became aware of Mike Shulman's answer [here](https://mathoverflow.net/questions/27219/proof-that-objects-are-colimits-of-generators?rq=1) which addresses the same question with the counterexample being the category of compact Hausdorff spaces. But is there a natural presentable example that one might expect to come across (some type of structured sets, for instance)? 
I'd like to get some intuition for this phenomenon. 



Can any object in a presentable category be written as a colimit of generators?
[Here](http://home.sandiego.edu/~shulman/papers/generators.pdf) is a short note by Mike Shulman with some relevant material. He remarks that the arrow category $\mathbb{2}$ inside $Cat$ (which of course is presentable) is a colimit generator in your sense (see his definition 3.6), but it is not a colimit-dense generator (see his definition 3.5) in the sense that any object of $C$ is a colimit of a functor into the full subcategory of $Cat$ containing $\mathbb{2}$. So this answers at least an implicit question of the post. 


I think I'll give him a poke, because he's undoubtedly better placed to handle this query. 


