What do I call a covariant functor which is a filtered colimit of representable functors?
I think the true confusion in Qiaochu's question never got resolved. I'll address the terminological issue raised by the first sentence of the question statement, but the second sentence of the question statement is not consistent with the first: there is only a tenuous connection between pro-objects in $C$ and presheaves on $C$, and also between pro-objects in $C$ and cofiltered limits of $\mathsf{Set}$-valued functors.
Let $C$ be a category. A pro-object in $C$ is a formal cofiltered limit in $C$. This can be represented by a functor $I \overset{F}{\to} C$ from a cofiltered category $I$. Or it can be represented as an object of the free completion of $C$, $[C,\mathsf{Set}]^\mathrm{op}$ which is the limit of $I \overset{F}{\to} C \overset{cY}{\to} [C,\mathsf{Set}]^\mathrm{op}$ (where $cY$ is the co-Yoneda embedding). The limit is taken in $[C,\mathsf{Set}]^\mathrm{op}$, so when we talk about it in terms of functors $C \to \mathsf{Set}$, it is actually a filtered colimit of corepresentable functors, not a cofiltered limit of representables.
This is reassuring on many grounds. Let me highlight a few:
The usage in the proceedings Qiaochu links to in the comments makes perfect sense.
Dually to pro-objects, an ind-object in $C$ is a formal filtered colimit in $C$, i.e. a filtered colimit of representables in $[C^\mathrm{op}, \mathsf{Set}]$. So $\mathrm{Pro}(C) = \mathrm{Ind}(C^\mathrm{op})^\mathrm{op}$. This is good; a formal cofiltered limit should certainly be a formal filtered colimit in the opposite category.
$\mathrm{Ind}(C)$ has lots of nice formal properties related to the fact that finite limits commute with filtered colimits in $\mathsf{Set}$. Since $\mathrm{Pro}(C)$ is just $\mathsf{Ind}$ in the opposite category, it should have all the same formal properties. But cofiltered limits in $\mathsf{Set}$ (or in a category of $\mathsf{Set}$-valued functors) don't have the same nice formal properties because $\mathsf{Set}$ isn't self-dual. So the dualization should not turn a filtered colimit of $\mathsf{Set}$-valued functors into a cofiltered limit of $\mathsf{Set}$-valued functors.
The temptation to call a cofiltered limit of representable functors $C^\mathrm{op} \to \mathsf{Set}$ arises, presumably, in analogy to something like "profinite group", where we often identify a object of $\mathrm{Pro}(\mathrm{FinGp})$ with its image along the functor $\mathrm{Pro}(\mathrm{FinGp}) \to \mathrm{TopGp}$ induced by the obvious functor $\mathrm{FinGp} \to \mathrm{TopGp}$ and the universal property of $\mathrm{Pro}$. By abuse of language, we call the image a "profinite group". Analogously, here we have a functor $\mathrm{Pro}(C) \to [C^\mathrm{op}, \mathsf{Set}]$ induced by the Yoneda embedding and the universal property of $\mathrm{Pro}$, and we want to think of a pro-object in $C$ in terms of its image along this functor.
This actually creates a pretty thorny terminological conundrum. I think you can get away with using the term "pro-representable presheaf on $C$" for a cofiltered limit of representable presheaves on $C$ as long as you're careful to distinguish this from "pro-representable copresheaf on $C^\mathrm{op}$", which should refer to an object of $\mathrm{Pro}(C^\mathrm{op})$, i.e. a filtered colimit of corepresentable copresheaves on $C^\mathrm{op}$, which is not the same thing.
It's basically a flat functor, which is sometimes called a (left) $\mathcal{C}$-torsor. Flat functors $F : \mathcal{C} \to \mathbf{Set}$ have the following elementary characterisation:
- There exists an object $C$ such that $F (C)$ is inhabited.
- Given $x \in F (C)$ and $y \in F (D)$, there exist an object $E$, an element $z \in F (E)$, and morphisms $E \to C$ and $E \to D$ such that $z$ is mapped to $x$ and $y$ (respectively).
- Given $x \in F (C)$, $y \in F (D)$, and a pair of morphisms $C \rightrightarrows D$ that both send $x$ to $y$, there exists an object $E$, and element $z \in F (E)$, and a morphism $E \to C$ such that $z$ is mapped to $x$.
In short, the comma category $(y \downarrow F)$, where $y : \mathcal{C}^\mathrm{op} \to [\mathcal{C}, \mathbf{Set}]$ is the Yoneda embedding, is filtered. Thus every flat functor occurs as a filtered colimit of representable functors (assuming $\mathcal{C}$ is small). The converse is more delicate, but amounts to showing that the category of flat functors is closed under filtered colimits in $[\mathcal{C}, \mathbf{Set}]$.