Martin (UAE Team Emirates) was faced with losing time on stage 14 to Mende when he punctured around the final climb within 5km from the finish.
The Irishman dropped from ninth to 10th overall as a result but his time deficit may be worse if not for Mitchelton-Scott team leader Yates, who is hunting for stage wins after surrendering his title hopes in the Alps.
Speaking before the start of stage 15, the pair explained the exchange that Martin first highlighted with a thank you on Twitter.
“He was out the arse after a puncture and he was coming up the climb solo,” Yates said. “I thought I’d give him a quick pull, like 500m. Why not, I’m not doing anything.
“Dan and I get along pretty well so thought I’d help him out a little bit. Not too much, because I need to save myself for today [stage 15] but do a little bit. One day he might return the favour, you never know do you.”
For Martin, the gesture was as much of a boost as the temporary pace Yates set.
“He didn’t have to do it - that’s the most important thing. I was grateful,” he said.
“I was in my own little world just trying to get to the top as fast as I could and having that little bit of encouragement helps a lot. It also helped having to weave through all the dropped riders, which is not exactly easy either.”
The Tour, following a handful of transition stages and the second rest day, moves to the Pyrenees and its most decisive point this coming week.
Yates is hopeful he has recovered enough from heat stress in the Alps to salvage his campaign and vie for a stage win.
“Pretty much every stage now kind of suits me unless it’s flat. There is about maybe four of five more opportunities and every chance we get we will try and do something,” he said.
“It’s still hot out here so still annoying but I’m feeling better every day so that’s the main thing.”
Martin, who finished sixth overall last season, will look to improve his placement on the overall standings. He’s had a mixed campaign of winning on the Mur de Bretagne, to suffering in crashes so far in this edition.
“We’re in a different space at the moment so I might not see much of him but if I get the chance to pay him back I’ll definitely help,” Martin said.
Stage 15 ended in Carcassonne with a great finish of the stage, and Christophe & Macca come up with a new conspiracy theory at Team Sky.