The SmallCategories Project aims to produce, maintain, and make available a database of small finite categories which is useful and useable.
The list of categories itself is produced by running the Minion constraint solver to enumerate all category structures on a given finite set, then eliminating isomorphic copies by brute force. You can find the source code for all of that here.
This website uses a postgres database to store the list of categories and selected propositions about them. The values of the propositions are computed using pycatlib, a tiny python library I wrote for this purpose.
You can find the source code for this website here.
SmallCategories is (so far) a personal project by myself, Ben Spitz. But if you're interested in contributing, please get in touch!