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A few typos: $(A, F_ A) \leq (A', F_{A'} )$ if $A \subseteq A'$ and $F_ A = F\_{A'} |\_{A \times \Delta ^ n}$" (not $F_ A = F_{A'} |_{K \times \Delta ^ n}$). We can assume $\tau$ carries the boundary $\partial\Delta^m$ into $A_{\mathrm{max}}$ (instead of $B$) Moreover, in the special case $k = 0$ (instead of $p = 0$). so that the composite map [...] corresponds to a $q$-cocartesian edge $e'$ of $X$. (missing article)

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A few typos:

[...] $(A, F_ A) \leq (A', F_{A'} )$ if $A \subseteq A'$ and $F_ A = F\_{A'} |\_{A \times \Delta ^ n}$ [...] (not $F_ A = F_{A'} |_{K \times \Delta ^ n}$).

We can assume $\tau$ carries the boundary $\partial\Delta^m$ into $A_{\mathrm{max}}$ [...] (instead of $B$)

Moreover, in the special case $k = 0$ [...] (instead of $p = 0$).

so that the composite map [...] corresponds to a $q$-cocartesian edge $e'$ of $X$. (missing article)

Yep. Thanks!

“We will complete the proof by showing that the lifting problem ... admit[s] a solution ...”

Also, could you make it more explicit that it suffices to solve the lifting problem in the case where $(A_{\mathrm{max}}, A')$ is replaced by $(\partial \Delta^m, \Delta^m)$?