UNKNOWN ACR ACFT AND A DC9 HAD A TCASII RA AT DTW.

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WE WERE ACR FLT XXX INBOUND TO DTW FROM DLH. IT WAS A CAVU DAY AND WE WERE CLRED FOR VISUAL RWY 21R BY TWR FREQ 135.00. ABOUT 4 MI OUT HE ASKED US IF WE COULD ACCEPT THE CTR RWY WHICH WE AGREED AND STARTED A SIDESTEP TO 21C. AFTER ESTABLISHED ON PROFILE RWY 21C THE TCASII YELLED 'TFC;' ILLUMINATED AN ACFT IN YELLOW RIGHT OVER US ON THE SCREEN. IT WAS DIFFICULT TRYING TO FIGURE OUT WHERE HE WAS. A COUPLE OF SECONDS LATER; TCASII RA OF 'DSND; DSND!' WE WERE TOO LOW TO DSND; SO I STARTED A BANK BACK TO THE R TOWARDS RWY 21R KNOWING NO ONE WAS OUT THERE. MY CAPT THEN LOOKED OUT HIS L WINDOW AND SLIGHTLY BEHIND US SAW ACR Y (DC9) BANK AWAY FROM US BACK TO THE E TOWARD RWY 21C. AT THAT INSTANT THE DC9 CREW ASKED TWR 118.4 'HEY; WHAT'S THAT ACR DOING ON THE CTR?' THE CTLR RESPONDED 'SORRY; HE WAS CLRED FOR THE CTR RWY FIRST; THAT WAS OUR MISTAKE.' BY THE WAY; PRIOR TO THIS TCASII RA WE WERE CLRED TO LAND BY THE TWR CTLR ON THE CTR RWY 21C (FREQ 118.4). MY BELIEF IS THAT THE 2 TWR CTLRS SIMPLY FORGOT TO COMMUNICATE TO EACH OTHER THAT THEY BOTH INTENDED ON CLRING THEIR ACFT TO STEP OVER TO THE RWY 21C.