On a topological space $X$, a net is defined as a map $A \ni \alpha \longmapsto x_{\alpha} \in X$ from a directed set $A$.

With this, compactness of $X$ (for instance) is equivalent to "every net $(x_\alpha)$ in $X$ has a subnet with a limit in $X$".

I wonder over what kind of set we quantify there. I guess it's not even a set, is it? So, how can we fix this? Does is suffice to consider equivalent classes of isomorphic directed sets up to the cardinality of the set $X$ -- which is hopefully a set? Is that kind of contemplation even necessary?