I am not completely satisfied by the accepted answer because the functor which characterizes the convergence of a filter depends on the limit. I therefore add another quite simple answer (written for sequences but this easily generalizes to filters and nets) to this old post.

The definition of a limit as a universal cone of a functor resembles the infimum (greatest lower bound) of a set in a very transparent way: Considering a partially ordered set $(X,\le)$ as a category with only one morphism from $x$ to $y$ if $x\le y$ and none otherwise, a subset $A$ of $X$ has an infimum if and only if the inclusion functor $A\hookrightarrow X$ has a limit. In particular, if the power set $\mathscr P(X)$ is ordered by inclusion the intersection of any subfamily $\mathscr A$ is a limit.

Let now $(x_n)_{n\in\mathbb N}$ be a sequence in some topological space $X$. Then the limit of the contravariant functor $F:\mathbb N\to \mathscr P(X)$ assigning to $n$ the set $F(n)=\overline{\{x_k:k\ge n\}}$ is the set of all limit points of the sequence.

I think that this is a strong relation between analytical and functorial limits although it does not yet characterize convergence of sequences. At least, if either $X$ is a compact Hausdorff space or $(x_n)_{n\in\mathbb N}$ is a Cauchy sequence in a Hausdorff uniform space, the sequence converges if and only if the set of limit points is a singleton.

5more comments