Continuity mistake: When they are leaving the first crime scene (the crashed UFO) in their Ford Taurus they are shown talking with headrests in the vehicle. They arrive at the headquarters for Zed's eulogy. After the eulogy they are back in the Ford Taurus and the headrests are gone. (00:07:40 - 00:09:50)
Continuity mistake: After K is falling with J from the rooftop, big chunks of concrete start to fall, yet disappear in some shots. When they land, the ground is clean, yet in one of the close-ups of K a chunk magically reappears next to his head. (00:17:50)
Continuity mistake: Boris' girlfriend is first shown to have a tattoo with "Boris" over her back. A few minutes later, after Boris says "I'm a bit of a romantic", the Lunar-Max guards walks over to the cake in Boris' girlfriend's hands and the tattoo on her back is shorter. (00:00:35 - 00:03:00)
Continuity mistake: At the Apollo rocket launch platform, J calls Boris the Animal twice, who lifts J up and launches him away. J glances at K, who's got both arms stretched out trying to save himself. A shot later, in the close-up, only his right arm is stretched out. (01:28:40)
Continuity mistake: When the Lunar-Max guards take Boris' girlfriend to visit his cell, one of the guards opens the door. Just as the shot finishes, the cell door can be seen open enough for two people to stand side by side. From the other side, there's only enough room for one person. (00:02:00)
Continuity mistake: When the huge alien fish throws J through the window of Wu's, J knocks over several rows of frogs hanging in the window as this happens. As he finishes being thrown out of the window, only one row of frogs is left. A few minutes later as K suspends J, K goes to exit Wu's and now there are suddenly two rows. (00:16:35 - 00:19:40)
Continuity mistake: When Boris' pet emerges from the cake, it kills the two guards. The Lunar-Max guard that had his hand stuffed in the cake has a spike put in his head and lands on the floor with his head resting against the cell wall. After the pet removes Boris' shackles, Boris goes to leave the cell and the guard with the cake on his hand is now lying on the floor almost entirely turned over. (00:03:15 - 00:04:00)
Answer: Earlier in the movie, the way K was talking about Boris was different than what J had come to expect of his partner. So J wanted to know what happened between Boris and K at Cape Canaveral that changed him. So more of what did Boris do to K rather than just the report of what K did.