I don't think there is enough information in your question. You have a field called Category that holds the value 'X', what is this fields type? Assuming that this field can hold more than the 1 character are there already records with both X and Y?
From what I understand of this question you could do something like this.
Update table
set category = category + "Y"
where table_id in (
select table_id
from table
where category like "X");
But this assumes that you have a database that does subqueries, and that you can do a concatination operation (you should be able to but you never know), and that you have a like operator.
WARNING: The above query is not a working query you are going to have to put it into the proper syntax for your database.
