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!
If $h:\Lambda^n_j\to X$ is an inner horn, then the image of $h$ is connected, and thus is contained in some connected component of $X$, which is in turn contained in one of the $X_s$. This means that $h$ actually factors through a map $h':\Lambda^n_j\to X_s$. Since $X_s$ is a quasicategory, $h'$ can be extended to $g':\Delta^n\to X_s$. Composing $g'$ with the inclusion $X_s\to X$ then gives an extension $g:\Delta^n\to X$ of $h$.