ACR RESPONDS TO TCASII RA. RECEIVES OPPOSITE DIRECTION RA.

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CAPT WAS FLYING. WE WERE CLBING. DEP CTL HAD CLRED US TO 13000 AND RPTED TFC AT 2 O'CLOCK; 14500; A COMMUTER. I TOLD HIM I WAS LOOKING. (WE WERE VERY LIGHT AND CLBING VERY FAST. WE WERE GOING THROUGH APPROX 12200 WHEN I TURNED MY ATTN OUTSIDE LOOKING FOR THE TFC). I DID SEE A PLANE; BUT WASN'T SURE IF IT WAS THE RIGHT ONE; SO I DIDN'T RPT IT IN SIGHT YET. ATC RPTED IT AGAIN; THEN I WAS DISTRACTED BY A 'TFC' ALERT FROM THE TCASII AND SHORTLY THEREAFTER AN 'ALT' ALERT. I LOOKED AND SAW OUR ALT OF 13300. I ALERTED THE CAPT; BUT HE SAID HE WAS FOLLOWING THE 'CLB' COMMAND FROM THE TCASII; WHICH I HAD MISSED. THEN; THE TCASII ISSUED A 'DSND' COMMAND WHEN WE REACHED APPROX 13500. THIS WAS INITIATED IMMEDIATELY BY THE CAPT AND WE LEVELED OFF AT OUR ASSIGNED ALT OF 13000. I TOLD ATC WE HAD BEEN AT 13500 DUE TO A TCASII ALERT AND WERE NOW BACK AT 13000. HE RESPONDED 'ROGER' AND THEN CLRED US TO 16000 MSL. AFTER DISCUSSION WITH ATC ON THE PHONE; WE SUSPECT THAT THE INITIAL CLB COMMAND MIGHT HAVE BEEN DUE TO TFC OTHER THAN THE COMMUTER AT 14500 AND THEN THE 'DSND' COMMAND WAS DUE TO HIM. AT THAT TIME (WHEN THE 'DSND' COMMAND WAS GIVEN). I DID HAVE POSITIVE CONTACT WITH THE COMMUTER AND NO CONFLICT WAS EVIDENT OR RPTED BY ATC.