you have a gts so you likely have a factory amp, that needs to be bypassed.
your best bet is to get a aftermarket headunit and wire that to your component set and other speakers, bypassing the amp. otherwise, i don't know what the wiring color chart is for a gts amp. but it appears that the amp splits the signals into high,mid,low.
that little battery type thing is most likely a passive crossover for the tweeter.
I can't give you any more help than that, sorry.