Problem:

Y adapter's different sequences to be annealed to the 5' and 3' ends of each molecule in a library.
The arms of the Y are unique, and the middle part, connected to the DNA fragment, is complementary.

Question: What are the advantages of this?

My take of this over having fully-complementary adapters (ADAPTER1 - - - - - ADAPTER1) is that:
-Upon primer annealing, the primer in turn would bind to the start AND end of the DNA fragment, and thus not allow synthesis to continue to until the end. Therefore, the next primer would not have a template to bind at the start of the fragment...
Y-shaped adapters also allow for paired-end sequencing. Fragments have a unique sequence on either end, which allows for the first "run" to sequence one side of all molecules, then synthesize the reverse and sequence that.

Am I on the right track? Please suggest your views.