I'm working on the following problem, from this link: https://faculty.math.illinois.edu/~rezk/quasicats.pdf
I will list the relevant definition and notation first.
And by $a_{ij}$, $a_{ij} = Cf(a)$, where $f \colon [1] \rightarrow [n]$, $f(0) = i, f(1) = j$.
Showing that if $C'$ is subcomplex then $f_{00}, f_{11} \in S$ for all $f \in S$ is easy, as $S \subseteq C_1$ and $C'_1$ is collection of $a \in C_1$ such that $a_{00}, a_{11} \in S$ (Note that $a_{01} = a$).
The reverse direction puzzles me, as it seems that $C'$ should commute with any simplicial operator as some map $<ij>$ composed with face or degeneracy map should still give some map of the form $<ab>$, but this means I didn't use the assumption that $f_{00}, f_{11} \in S$ for all $f \in S$!
And I have no idea about the other problem of $C'$ being a subcategory if and only if for all $u \in C_2$ we have $u_{01}, u_{12} \in S$ implies $u_{02} \in S$, so I would appreciate if you could point me in the right direction for that.
Thanks!