nLab stable pro-object

Given any category $C$, we can form the corresponding category of pro-objects in $C$, which is denoted by $pro$-$C$. Since the category $\mathbb{0}$ with one morphism is a coflitered category, within $pro$-$C$, we have all pro-objects of the form $X: \mathbb{0}\to C$. Clearly such a functor is completely determined by the single object, $X(*)$, of $C$ to which it corresponds. This gives a functor:

$$c: C\to pro-C$$

which embeds the category $C$ in $pro$-$C$. (This is really the Yoneda embedding in disguise.)

Definition

Any pro-object isomorphic in $pro$-$C$ to one of the form, $c(X)$, for $X$ an object of $C$, is called stable or essentially constant.

Stability problem

In any given categorical context, the so-called stability problem is the problem of deciding what internal criteria can be applied to check if a given pro-object in that context, is or is not stable,

If $C$ is an abelian category, it is relatively simple to give necessary and sufficient ‘internal’ conditions for a given pro-object to be essentially constant. It must be both essentially epimorphic and essentially monomorphic?.