You don't catch enemies with the Trainer dressphere. The only thing that sounds remotely similar is learning Blue Magic, but that's the Gun Mage DS. My apologies, I digress. The Trainer DS enables each girl to battle with the help of their personal pet: Yuna has Kogoro (dog), Rikku has Ghiki (monkey) and Paine has Flurry (pheasant, apparently). The stats vary slightly between the girls, as do the abilities they learn, the most important of these being HP Stroll, MP Stroll and Half MP.