Popovich still isn’t sure what he has roster-wise, but he can point to at least one luxury as he finally settles into his dream job as Mike Krzyzewski’s successor: Mitchell’s rise alongside the All-Star guard Kemba Walker will enable the United States to field an elite starting backcourt at the FIBA World Cup in China from Aug. 31 to Sept. 15.

“Just try to throw yourself completely into it,” Snyder said he told Mitchell. “And try to communicate with Coach as much as you can. Be a sponge.”

Snyder understands the benefits of a stint at the School of Pop as well as anyone. He suddenly found himself in hoops limbo in 2006 after a messy end to seven seasons at Missouri, then found a path to a rebirth in the N.B.A. by joining the Spurs for the 2007-8 season as the coach of their minor league affiliate in Austin, Tex.

As legend has it, Snyder was invited to a Spurs playoff planning meeting after his first season coaching the Austin Toros and made enough of an impression to convince Popovich that he needed to be at every subsequent meeting — although Snyder doesn’t exactly remember it that way.

“I just tried to be seen and not heard,” he said. “I do recall at one point probably being a little reluctant to talk. I think I said something to myself under my breath instead of writing it down for the group, and that was my moment — it was the first time Pop got on me and said, ‘O.K., Quin, if you have something to say, say it.’”

Snyder gradually learned to voice his opinion, then proceeded to assistant coaching jobs with the Philadelphia 76ers, the Los Angeles Lakers and the Atlanta Hawks — as well as CSKA Moscow in Russia under the Italian coach Ettore Messina. Snyder landed his first head coaching job with Utah entering the 2014-15 season when he was hired by his fellow Spurs alumnus Dennis Lindsey.