Let $\mathcal{C}$ be a category. The pro-category pro-$\mathcal{C}$ is defined as (see this [nLab](https://ncatlab.org/nlab/show/pro-object) page) follows: its objects are diagrams $F: D\to \mathcal{C}$ where $D$ is a small cofiltered category. The Hom set between $F: D\to \mathcal{C}$ and $G: E\to \mathcal{C}$ is given by
$$
\text{pro-}\mathcal{C}(F,G):=lim\_{e\in E}colim\_{d\in D}(Fd,Ge).
$$


My question is:



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Why do we need cofiltered condition on the index category in the definition of pro-categories?
The introduction of $Pro(C)$ is in [Grothendieck's seminar Bourbaki 195, (1960)](http://archive.numdam.org/article/SB_1958-1960__5__369_0.pdf). There you can see the properties that using pro-objects gives you. This is related to Dylan's two comments above. In the classical case of handling inverse systems (pro-objects) of fin. dim. vector spaces or finite groups, then the limit of the pro-object is compact in a nice way and you can rebuild the inverse system from it. Without the cofiltering condition you do not get that. Likewise, the classical theory of Cech homology naturally gives rise to inverse systems and not just general functors. We thus have a situation in which several important examples / applications lead to such pro-representable functors AND such functors are better behaved.... It therefore looks to be a GOOD THING to restrict attention to pro-representable functors in these contexts. (N.B. I am missing many other situations where a naturally constructed functor is pro-representable.)


Another N.B. is that there are other forms of pro-representability involving more general properties than the ones involved in pro-representability *per se*, but going to absolutely general functors / presheaves, (with no conditions) usually involves other methods.


