Both Kakashi and Obito awaken their Mangekyou at the same time as told in this question When did Kakashi and Obito awaken their Mangekyou Sharingan? The sharingan Kakashi owns originally came from Obito, so is it due to Obito also being there at the moment Rin died that Kakashi's Mangekyou also awakend? Or asked in a different way. Would Kakashi have been able to awaken the Mangekyou without Obito being there?
Interesting question with no definite answer, so I'll post my 2c here.
The Mangekyou Sharingan is activated by the immense grief caused by killing the person closest to you. However, it seems like you don't actually have to kill him, but witnessing his death is enough.
For example, Itachi did not actually kill Shisui, rather, he watched him commit suicide (filler).
I believe that both of them awakened their Mangekyou Sharingans separately, and because they were both at the scene when it happened, it happened simultaneously.
The Sharingan was stated to be the result of special chakra inside a Uchiha's brain altering their optic nerve when they lose those they love as Kakashi is not a Uchiha he is unlikely to be able to produce the special chakra required to gain the Mangekyou Sharingan.
Obito could still see through Kakashi's Sharingan when in close proximity had showing he is still connected to the Sharingan he gave Kakashi so I believe that through that connection the special chakra produced in Obito brain evolving his Sharingan into the Mangekyou Sharingan was trasfered to Kakashi's Sharingan causing it to evolve into the Mangekyou Sharingan.
The visual prowess of the eye does not depend on the will of the person using it but on the one to whom it originally belongs to (because of Madara using his brother's visual prowess and even Sasuke using Itachi's visual prowess without originally having the abilities. Also, Itachi used Shisui's visual prowess without having Kotoamatsukami originally). Therefore, Kakashi did not awaken the Mangekyou. It was Obito who awakened it in both eyes.