@Valter

>\\(f_{\ast}(A)\\) does not just give you "all the buckets that only contain balls from A", but also all the buckets that never get any ball from \\(X\\).

Yes that's what I thought after I drew the diagrams! The word "all" was getting to me because by my mortal logic where all excludes the vacuous truth that John talks about in the comments below, this would lead to left and right adjoints being the same map for all of the examples above but it isn't.