Kind of, though Haskell's monads are much more specific than those of category theory.

A tuple/triple of functions of certain signatures?)

No, a Category k a b such that k a b is isomorphic to a -> k () b.

A Haskell Category is again more specific than a general category though!

http://web.jaguarpaw.co.uk/~tom/blog/posts/2012-09-02-what-i...