What are the circumstances of what Obito the Sharingan used against Konan and the Kakashi's Sharingan? Obito lost a Sharingan after using Izanagi against Konan. And now its shown that he gave Mangekyo Sharingan to Kakashi in both his eyes. What's the connection?
If you have seen the fight between Tobi (Obito) and Konan, he actually used his left eye which he had given to Kakashi. So the Sharingan he used for Izanagi was of the other (Uchiha). 



Also, Obito gave Kakashi his Sharingan (the right one) and Kakashi already has the left Sharingan, so he has both and talking about Mangekyo, Kakashi has already awakened that power.