Consider for simplicity the category of pointed sets $Set_*$. Let's call an object $G$ of a pointed category $C$ a generator if for every nonzero object $X \in C$ there is a nonzero map $G \to X$. In the category of pointed sets, for example, the two-point set is a generator (and a cogenerator as well).
Does the category $Pro-Set_*$ of cofiltered diagrams of pointed sets have a generator as well? If $X$ is a pro-pointed set with nontrivial limit then there is a nontrivial map $G \to X$, where $G$ is a generator of pointed sets, considered as a one-object diagram. However, there are many pro-sets with trivial limit. For example, the pro-set $X\colon \mathbb N \to Set$ given by $X(i) = \mathbb N$ and $X(i+1 \to i)(n) = n+1$ is a pro-set with empty limit, but which is nontrivial -- add a point in every degree to get an example in pointed sets.
Thus, my question is: Does the category of pro-pointed sets have a generator?
And, if the answer was yes, does more generally $Pro-C$ have a generator whenever $C$ has one?