Set a short while after the events of Sun and Moon. I see both professors as in their late twenties, early thirties. Guzma's maybe early to mid-twenties. Guzma's trying to make up for being a shit in the game, and he's working with Kukui now, rather than against him. It's a blurb, I'm sorry; I was bitten by the tiniest plot bunny.
"Absolutely fascinating," Professor Sycamore had said, for perhaps the twentieth time since he set foot on Melemele Island. He poured over Kukui's research, enamored by both his notes on Z-Crystals and the region's unique variant Pokémon. Honestly, an ice type Vulpix! He wasn't going to leave until he had one in his possession - they were just too darling.
"Hala will be doing a demonstration this evening during the festival that you may not want to miss," Kukui said with a smirk. He'd given the awestruck professor a tour of his side of the island earlier in the day, amused by the other man's constant amazement.
"Of course, of course," Sycamore responded distractedly, rolling his sleeves up to his elbows. He'd long since abandoned his lab coat due to the overbearing heat of the island. "Wait, what sort of demonstration?"
"Oh, a little of this. A little of that. He's in possession of several Z Crystals, y'know?" His smirk grew into a knowing grin as Sycamore turned his full attention towards him, face showing his barely restrained excitement. He was like a child set loose in a toy store for the very first time.
"Maybe we can do some shopping beforehand, find you something better suited for the climate, yeah?"
"Certainly," he reached up to pop the third button down on his deep purple shirt, trying to expose more of his overheated skin to the fan Kukui had turned on for his benefit. "I'd known it was to be warm here, but I wasn't expecting-"
"Oy, Kukui, you down here?" Footsteps thundered down the stairs in Kukui's lab, revealing a pale young man with a shock of white hair. "Shoulda known, ya big nerd. Always got your nose in them boo- oh."
The intruder's noticeable slump straightened when he caught site of Kukui's guest, his face gone slack in surprise. "Uh…"
"Guzma, this is Professor Sycamore. He's visiting from Kalos. Augustine, this is our resident scab, Guzma. Feel free to ignore the little runt. He only works for me part time."
Either not hearing Kukui's jabs, or because of his proper upbringing, Sycamore only turned a pleasant smile Guzma's way. "Charmed," he said, fanning his shirt to invite more cool air inside. Guzma watched as a bead of sweat trailed over his collarbone and disappeared under his shirt, his mouth suddenly too dry for words. Kukui watched their interaction, tamping down on the sudden desire to burst out laughing.
"Guzma's charmed too, I bet. Is there something you needed, Skull-Kid?"
Guzma blinked, snapping back to himself and glancing Kukui's way. "N-uh, I'll wait. Upstairs. I'll be upstairs," he managed, turning heel and stumbling back up the way he came.
"Absolutely fascinating," Sycamore murmured, watching him go.