My motivation for this question is pure abstract nonsense, but I'd like to define continuous maps or continuity itself in a functorial/categorical way, since every continuous map induces a map (of sets) between the topologies in the opposite direcction.
Therefore, I could say that a continuous map $f:X\to Y$ is just a map of sets, which induces a map $f^*:Top(Y)\to Top(X)$. The thing is that given a set, there are lots of topologies on it and given two topological spaces there are in general plenty of continuous maps, so I've thought of taking a subcategory $S$ of Set$\times \mathbf{Set}^{op}$ whose pairs consist of a set and a topology on it. In this setting, I consider a functor $F:S\to \mathbf{Set}$ which is the second projection on objets, and sending each arrow $(x,t(x))\to (y,t(y))$ to an arrow $t(y)\to t(x)$. In the best case this would give a class of continuous maps $x\to y$, but I'm not sure that every map $t(y)\to t(x)$ gives rise to a well-defined map $x\to y$.
Are there definitions of continuity or other topological concepts following this approach?