Despite being Rivals of their own design, or perhaps only Gai's design, Kakashi and Gai really knew very little about one another other than by reputation and their infrequent meetings. They were, at best, impersonal, and it was a conscious decision on their part not to know much more than what mattered most.
So, on their first meeting after Team Minato was reborn, Gai officially introduced himself at last, bow and all, and waited in anxious silence for Kakashi to do the same. It was, Gai proclaimed, what one did when meeting teammates. The other boy, however, did not do formalities. He drank in Gai with one cold look and then turned away to contemplate the stone memorial instead. As Kakashi took his time tracing with his fingers the name of Uchiha Obito, Gai struggled for something to keep his Rival's attention, something they had in common other than challenges and their current teammates, and ended up recounting his own Chunin exam. He babbled.
But Kakashi was unimpressed and made that known; he had been present and knew quite well that Gai had fought Obito in the final section of the exam—the Uchiha and the green-clad boy had been teammates, along with Rin, for the duration of that exam, since Kakashi had already completed it years before. To shut down this line of conversation, the Copy Ninja coolly asked why Gai's original team hadn't taken the exam with him.
They did not want to, apparently and according to Gai, so he was hand-picked—a fact he was clearly proud of—by Minato-sensei himself to replace Kakashi. He replaced no one, Kakashi coldly informed him. Gai nodded, and said he understood his place.
After another awkward silence, he quietly told Kakashi how he had known Obito from the academy. They had sat next to each other during their last year. Obito liked to doodle on his homework and had difficulty, at first, throwing shuriken. He was a nice kid, though, and had seemed excited to have the great Copy-Nin for a teammate. Kakashi said nothing; he perhaps shrugged once in reply before he turned back to the memorial stone. Gai fell into silence.
It wasn't until nearly thirty minutes later, after Minato-sensei and Rin had come, spoken to them, and left for the ramen stand, that one of the two boys made another attempt to converse. Out of the empty air, Kakashi quietly asked if Obito was such a crybaby back then, too. Gai nodded, and recalled having once accidently punched Obito in the nose during a routine lesson in Taijutsu. Obito had cried for a whole ten minutes and Gai had to help take him to the infirmary. He had a bloody nose, and claimed to have had something in his eyes. Allergies, Gai said.
Kakashi teased that Obito was allergic to Gai…Gai, in turn, was too affronted to realize that the other had just made a joke. When Kakashi informed Gai that Obito didn't especially like this most recent rivals' challenge, Gai didn't understand. Kakashi touched his own masked nose and simply said that Obito wanted to cry.
With profuse apology, and a wee bit of skepticism, Gai promised not to punch his Rival in the nose again, for Obito's sake, since he was allergic to such things. Maybe Obito was grateful. If he was, Kakashi didn't say.