ACR LEAVING ZOB AIRSPACE INTO ZNY AREA WITHOUT COORD OF NEW ALT ASSIGNMENT. OPERATIONAL DEVIATION.

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I AM AN FPL CTLR CONDUCTING RADAR OJT ON A LOW ALT SECTOR. THE DEVELOPMENTAL CTLR IN TRNING HAS OVER 100 HRS OJT LOGGED ON THIS POS; AND ANOTHER 100 HRS SPLIT BTWN 2 OTHER RADAR SECTORS WITH NO RADAR SECTOR CERTIFICATIONS. AT THIS POINT IN TRNING HRS MOST DEVELOPMENTAL CTLRS HAVE CHKED OUT AT THIS SECTOR; CONSIDERED THE EASIEST SECTOR IN OUR AREA. ACR X AT FL210 RPTS LIGHT TO MODERATE CHOP AND REQUESTS A DSCNT TO 17000'. ACR X IS 15 MI W OF THE CLE/NY CENTER BOUNDARY HDG E. THE ACFT IS IN HDOF STATUS TO ZNY SECTOR 75; A HIGH LEVEL (FL180-FL600) SECTOR. THE UNDERLYING SECTOR TYS CONTROLS 8000-17000' MSL. THE RADAR ASSOCIATE (D-SIDE); ALSO A TRNEE RECEIVING OJT INSTRUCTION; CALLS ZNY 75 AND STATES ACR X REQUEST FOR 17000'. ZNY RESPONDS 'I HAVE HIS REQUEST.' THE RADAR TRNEE THEN CALLS ZNY 75 AND STATES; 'ACR X WANTS 17000'' AND ZNY ANSWERS; 'YEAH; I GOT THAT.' THE RADAR TRNEE TAKES BACK THE HDOF; ASSIGNS 17000' TO ACR X; UPDATES THE ASSIGNED ALT IN THE DATA BASE AND MANUALLY ADDRESSES THE HDOF TO SECTOR 75. THE ACFT IS 10 MI INSIDE ZNY; FL200 DSNDING BEFORE HDOF IS FINALLY ACCOMPLISHED. 1) SIMULTANEOUS OJT AT BOTH THE RADAR AND ASSOCIATE RADAR POSITIONS. 2) LACK OF STANDARD PHRASEOLOGY LED TO MISUNDERSTANDING; 'YEAH; I GOT THAT;' RATHER THEN APPROVED; UNABLE; ETC. 3) MOST IMPORTANTLY; THERE IS A TENDENCY TO LET A TRNEE GO AS FAR AS POSSIBLE IN TRNING TO MAXIMIZE THE BENEFITS OF THE TRNING SITUATION. I WAS NOT SPECIFIC ENOUGH WHEN I ASKED THE TRNEE; 'ARE YOU JUST GOING TO LET ACR X GO?' AS AN FPL; I AM LOOKING AT AN ACFT DSNDING THROUGH A SECTOR 75 ALT TO A SECTOR TYS ALT IN HDOF TO SECTOR 75 5 MI W OF THE BOUNDARY. THE RADAR TRNEE POINTED THE ACFT OUT TO ANOTHER ZOB SECTOR RATHER THAN FOLLOW THROUGH WITH THE HDOF. I DIRECTED THE ASSOCIATE CTLR TRNEE TO HDOF THE ACFT FO ZNY; BUT THE COORD WAS NOT TIMELY. THERE MUST BE AN OVERIDING DUTY OF PROVIDING A SVC TO OUR CUSTOMERS (SYS USERS) OF SAFE; ORDERLY AND EXPEDITIOUS FLOW OF TFC; AND MAX TRNING BENEFIT MUST BE SUBORDINATE TO THAT DUTY!