Did Obito let Kakashi win? So, during Obito and Kakashi's fight in the Kamui dimension, Obito gets a hole in his chest. After watching the episode, most of the viewers commented that Obito "was defeated" even though he had the Rinnegan. But wasn't it because Obito "let" Kakashi win to undo Madara's Seal (as we see in chapter 675, Obito gets to know about the seal and wishes to undo it) and become the Ten Tails Jinchuuriki? Also, I wanted to know that would Kakashi (a non-Uchiha and a non-Senju) have been able to actually defeat Obito? Because during their fight, Obito didn't use Izanagi or even one Rinnegan ability (even though, just moments later, he seals the Ten Tails inside him with the Rinnegan). I am compelled to say this, but Obito's abilities are no joke.

Please provide a convincing answer to explain exactly what happened during the fight and what would've happened if Obito was really serious.
Yes he did. Though It was not completely intentional. Specifically, he need a hole through his heart, because Madara placed a Seal on his heart, which he planned on using to control him. Obito later noted that He could not become the 10 tailed Jinchuriki because of that Seal. So Obito forced Kakashi use his Raikiri to pierce his Heart, destroying the seal. However, it was not a simple, let him do it kind of deal.

The Wiki unfortunately leaves out most of these details, but the Curse Mark's Entry on the Wiki has some of them, Its called The Forbidden Individual Cursed Tag. It forced Obito to not betray Madara, so in that respect, Obito could not just let it happen. However, he could indirectly let it happen, by forcing it, and otherwise weakening himself so that he could not stop it. In the Kamui Dimention, He could not use his intangibility, which would have been his main and possibly only solid defense against Raikiri. He also restricted his Chakra useage, saving it for the future battles in the war, rather then using it to win against kakashi, as he used almost no ninjutsu, and almost no Rinnegan techniques.