# category theory – The coproduct of any indexed collection of quasicategories is a quasicategory.

I’m reading the following document: https://faculty.math.illinois.edu/~rezk/quasicats.pdf

The proposition I’m having trouble with is the following: The coproduct of any indexed collection of quasicategories is a quasicategory. (6.7 in the document)

To prove this proposition, the author uses the following information.

Given this information, the proof goes as follows:

I understand everything upto “The proof is now straightforward…”. That is, I don’t know how to use 6.12 and 6.11 to reach the conclusion.

At this point, I’m not even sure how contentedness would imply that inner horn filling condition is fulfilled.

Any help on understanding the proof would be appreciated. Thanks!