It's not like the whole fight was a lie. Kakashi did smack him round a couple times, and honestly without kamui Kakashi just just overall more skilled. I don't think Obito let him win, there are plenty indications that show that Obito was actually trying. Even Obito trying to defend himself from and stabbing Kakashi in the end, the way kishi drew the fight to Mirror their past battle and the fact that they both indicated the seal of confrontation before the battle. Obito trying to confuse Kakashi with multiple genjutsu. I think in the end Obito realised he was losing the battle and needed a way out in order to fulfil his plans. He even said Kakashi won the fight but won't let him win the war.
And before anyone says he didn't use kamui intangibility, giant fireballs or wood release. It's clear that Obito cannot use kamui to the fullest capabilities when in the other dimension (meaning no intangibility), his bomb blast dance is enhanced with kamui as well which He probably couldn't use in the dimension either, and Obitos wood release isn't even better than yamatos, nothing Kakashi wouldn't deal with. Kakashi didn't use many jutsu either, could use kamui as well.
Besides, no where does the scan say, I let Kakashi win. If I see a scan of Obito saying I allowed Kakashi to win the fight then I'll say otherwise but I know that scan doesn't exist.