Flight Crew Commercial; Flight Crew Flight Instructor; Flight Crew Instrument
Flight Crew Last 90 Days 90; Flight Crew Total 5200; Flight Crew Type 3900
Situational Awareness
ATC Issue All Types; Conflict NMAC
Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Airspace Structure; Human Factors; Procedure
I was beginning a dual instruction flight from [the airport] with a requested departure to the East. We had climbed through 800 FT and were in our turn to the east. Traffic inbound had been previously reported to us by ATC before takeoff and an update to its position given to us while we were turning on course. We had been likewise reported to the other aircraft; Aircraft Y; located to the northeast of the airport; upon our departure and upon our turn. Aircraft Y reported us in site and stated he would avoid us. ATC instructed us to 'climb; climb' and we did. We were watching for the traffic while departing but did not see them until after they had passed below us. We did initiate the turn on course as approved because I believed the traffic to be further away from us. The traffic was in a challenging position as we were in a climbing right turn from [the runway] to the east and they were approaching from the northeast. My flight was a training mission with a student pilot in the left seat.  All information was coordinated by and given to both parties. My concern is that the clearance to depart and turn created a conflict and that both aircraft were allowed to get so close. There had been several other conflicts in the airspace earlier in the day. I had just had discussion with another instructor about aircraft being mistakenly identified to each other in the traffic pattern.
A PA28 instructor pilot reported that during climb they had a near miss with another PA28 that was on descent.