Gojo, as it turns out, went shopping for sweets and petty, cheap trinkets that look like they absolutely came from gacha machines. He was thoughtful enough to buy quite a lot of both, so if they're feeling hungry, or if there are any collectable keychains they'd like, he has a selection that he will deign to let them take from!

"...I am not going to ask."

"Me neither but I am going to snack, I am starving, that was very energy-consuming."

"If sweets are on offer I will not turn them down," agrees Hayashi.

Gojo magnanimously hands over some of his sweets to his students. ... Not all of them, though. Some of them are definitely his. But they are very good, very expensive sweets.

"Look, I got a set of little tiny critters that hang from glasses! I'm going to give some of them to Maki-chan, I expect she'll be thrilled."

"Maki-chan?" he says with his mouth full.

"Zen'in Maki-senpai is a second-year from Jujutsu Tech."

"She has physical prowess that almost rivals yours! And no cursed energy to speak of. Wears a pair of cursed glasses that let her see curses at all." He demonstrably makes circles with fingers over his eyes, because clearly this point needed illustrating. "Very talented, and she makes the other school so mad."

Hayashi has the good manners to wait until her mouth is no longer full to say, "She what?"

"Oh, yes, the Zen'in clan has been against her career, blocking her promotion and everything. They're very mad that she has been learning here! That," he points at Itadori's cleaver, "is hers, actually."

"That wasn't my - okay. I am impressed by Zen'in-senpai's competence and now kind of want to meet her."

"...I am, technically, from the Zen'in clan. I wasn't raised by them, though."

Oh gosh, backstory unlocked!

"That sounds like a complicated situation," she says, doing her very best to be polite and not immediately start interrogating him for details. "But I think whether or not you are from a specific clan is up to you, not. Just bloodline."

"And you see what name I use."

"I do! And it is much less confusing to only have one Zen'in in the school." Pause. ".... There isn't another, right, I didn't get a student roster to read over or anything."

"Zen'in Maki-senpai's twin Mai is enrolled at the Kyoto school."

"Oh. But she's over in Kyoto, so it probably won't come up."

"Probably not."

"There is actually a yearly goodwill competition event between the sister schools, to foster good relations! But it's for second and third years only, so. Barring unusual circumstances, it'll be over a year!"

"Oh. Okay."

"We going to meet our upperclassmen soon?"

"They're away handling the last of the winter blues curses, but they'll be back within the next month. Our profession is kind of seasonal, changes with how the population's doing. Summer's usually our down time, it's why we teach classes then."

"...yet another thing that makes sense and I did not expect. Cool."

The following days are rather quiet. Gojo is absent over half of the time, leaving them to teach each other and figure things out on their own. When he is there, he's helpful in his typical idiosyncratic way. Itadori graduates from a sheep that makes noise to an elephant whose trunk uncurls proportional to the amount of cursed energy channeled into it, then eventually a metronome to go with it, to add practicing timing to the mix. Hayashi gets a promise that he'll make a request for something like fishing line-but-stronger, and is then told to go practice her sympathy technique on things she finds in the woods, or in an urban environment, instead of just what she always has with her. Fushiguro gets held back from what few and minor curse exorcisms occur, and he and Hayashi are gently prodded to start learning some close combat techniques from Itadori.

And then, almost two weeks after Hayashi's arrival, he has to go.

"I'll be away for a few days, maybe a week at most," he assures. "There are a couple of scattered curses up north that everybody's decided just need my personal attention. The downsides of being the strongest. Don't do anything I wouldn't do, and don't take my absence as an excuse to be lazy!"

"Sensei, most of our education is self study anyway. We'll hardly notice you're gone," quips Hayashi, lightly.