As I mentioned in my comment below: https://news.ycombinator.com/item?id=10529838 , the category of SQL queries has coproducts and products, but the category of SQL tables does not. Coproduct and product tables are very useful in their own right.
Not sure what definition you're using for the category of tables, but I don't think the distinction between table and query is really that significant, at least from a theory point of view. You can declare views and materialized views. In some SQL dialects you can even define triggers which allow you to 'update' them, not that I would have thought mutability would be particularly nice to reason about in a category-theoretic framework.
If you really want to do theory on this stuff, just use the relational algebra, or better yet just plain first-order logic. Much nicer, you have all the products and coproducts you want, and the results can probably be re-applied to SQL with a bit of cludge-work :)
Agreed that SQL is ugly as hell, but if you want to talk about its theoretical properties that's a separate debate. Theory doesn't care whether something's aesthetically pleasing, just whether it's possible.