Let $\mathscr{D}$ be a category, where objects are directed posets and morphisms are cofinal maps. We may also consider $\mathscr{D}_m$, where morphisms, in addition, monotone.
Does $\mathscr{D}$ (or $\mathscr{D}_m$) have inverse limits? If no, is there a reasonable definition of category of directed posets which will have?
I came to this question when thinking of Tychonoff theorem about compactness. I wanted to prove it (for fun), by generalizing diagonal [subsequence] argument to subnets, and thought the notion of inverse limit would be useful. Informally, if we construct the subnets carefully, the cofinal maps from the definition of subnet make the collection of index posets of out nets directed system.
I know not much in category theory, and it isn't trivial for me whether the inverse limit exists.
The categories $\mathscr{D}$ and $\mathscr{D}_m$ do not have inverse limits. For instance, consider the inverse system $A_n=\{m\in\mathbb{N}:m\geq n\}$, with the maps being the inclusion maps. If $B$ is any object which maps to the inverse system (in either $\mathscr{D}$ or $\mathscr{D}_m$), the image of the map $B\to A_0=\mathbb{N}$ would have to be contained in $A_n$ for all $n$. But $\bigcap A_n=\emptyset$, so this implies $B$ is empty. Since the empty poset is not directed, this means that no object can map to the inverse system. In particular, it cannot have an inverse limit.
To avoid this difficulty, you might try to modify the category so that in this inverse system, the inclusion maps $A_{n+1}\to A_n$ are isomorphisms. For instance, maybe you should impose an equivalence relation on the morphisms, saying that two morphisms are the same if they eventually are equal. I'm not sure whether you can get a better-behaved category by doing something like this.