Double A and Triple A are basically the "performance" levels of the minor leagues. High A ball, A ball, rookie league are the "development" levels. The line is a bit blurred with some on the double A level, but pretty much if you show you can handle Double A pretty easily, the challenge is not there for you to make any sort of impression.
So you'll get a call-up to either triple A or the majors from double A if you "perform". J.R. House last year started in double A. He whacked the ball around for pretty much the majority of the season in Corpus and got sent to Round Rock some time in August (may of been late July). He whacked the ball around in triple A too, still not displaying a penchant for playing any position that well, but the guy was hitting.
So he "performed" and was rewarded with a call up to the majors. As such, he got a chance to sign a major league contract this offseason with the Baltimore Orioles. As long as you have the designated hitter league, a hitter... be he a Double A or a Triple A (doesn't matter)... that shows he can flat out hit, he then becomes a very good candidate for DH needs in that league.
So, J.R. House had no trouble starting in Double A and turning it in to a major league gig. Nobody else should either. It's not the Houston Astros who need to do right by Todd Self, it is Self himself who should take this chance and run with it... like, well... J.R. House.