NMAC.

Narrative

DURING A TRAINING FLT THAT WAS A CHKOUT IN AN SMA X; WE WERE ENRTE FROM NAPA ARPT (APC) TO CONCORD (CCR) ARPT. I HAD ASKED THE PLT BEING CHKED OUT TO CLB TO 2500 MSL AND PROCEED TO CCR ARPT. AFTER LEVEL OFF; CCR ATIS WAS RECEIVED AND THEY WERE LNDG RWY 32. VISIBILITY WAS RPTED AS 4 MI IN HAZE. ON INITIAL CONTACT TO THE TWR (CCR) THE 'STUDENT' RPTED 8 MI N OF THE ARPT WITH ATIS INFO USING THE CALL SIGN NXXXX. OUR FULL TAIL NUMBER WAS NXXXXT. THE TWR ASKED IF NXXXX WAS OUR FULL CALL SIGN. THE STUDENT RESPONDED AFFIRMATIVE NXXXX. AT THAT TIME I TOLD THE STUDENT TO TELL THE TWR (CCR) OUR FULL CALL SIGN. HE DID NOT. WE WERE APCHING THE ARPT AT 120 KIAS AND WERE GETTING THERE RAPIDLY. THE STUDENT HAD STARTED HDG SWESTERLY AFTER THE INITIAL CALL WHEN TWR HAD ASSIGNED R TFC RWY 32R; RPT DOWNWIND. WHEN WE SPOTTED THE ARPT WE WERE 2 MI NW OF RWY 32L/R. I CALLED THE TWR AT THAT POINT AND INFORMED THEM WE WERE IN A BETTER POS FOR L TFC RWY 32L. THERE WAS CONFUSION AT THAT POINT BECAUSE HE THOUGHT OUR CALL SIGN WAS XXXX AND WHEN I CALLED HIM AS XXXXT HE ASKED ME TO CONFIRM WHO AND WHERE WE WERE. I DID; AND HE APPROVED L TFC RWY 32L AND AS WE DID THAT; WE PASSED ABOVE AND TO THE R OF AN SMA Y. TWR THEN HAD US IN SIGHT AND ASKED IF WE HAD THE SMA Y IN SIGHT AND WE DID THE WHOLE TIME. TWR TOLD US TO FOLLOW AN SMA Z AND ASKED THE SMA Y IF HE SAW US; HE DID. THE TWR SEEMED ANNOYED AT US BECAUSE OF THE IMPROPER CALL SIGN AND POS IN REF TO THE ARPT. THE REASON I AM SUBMITTING THIS RPT IS BECAUSE OF THE SAFETY FACTORS INVOLVED WITH A HIGH PERFORMANCE ACFT CHKOUT; LOW PATTERN VISIBILITY; IMPROPER CALL SIGN; DUAL INSTRUCTION. WHEN DOING CHKOUTS OR ANY KIND OF INSTRUCTION IT IS VERY IMPORTANT TO DIVIDE YOUR ATTN BTWN THE ACFT; AIRSPACE; AND STUDENT AND IF THERE IS A BREAKDOWN IN ANY AREA IT CAN SNOWBALL INSTANTLY INTO AN UNSAFE SIT WHICH THIS WAS LEADING TO. ANY TIME WE FLY WE MUST MAINTAIN VIGILANCE AND NOT EXCEED OUR OWN PERSONAL LIMITATIONS IN ANY OP.