Understanding which pet to put battle/Morphing

I have been playing awhile now. But I just noticed something.So when you choose which pet to put into battle/fighting and then u can choose which pet you can put into morphing. Do u go bye the pets skill that u put into battle to help you? Like for example say I put cinder in battle when u click on his skill. It shows another pet I guess that helps cinder in battle. Is that the pet you want to put in morphing? I hope this wasn't confusing. Just trying to completely understand it and make sure I am doing it right is all. Or does it really matter what pet u put into battle and morphing?