Proof of Proposition 4.5.6.23.
Let $\mathscr {F}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram which satisfies condition $(\ast _ n)$ of Proposition 4.5.6.23 for every integer $n \geq 0$; we wish to show that $\mathscr {F}$ is isofibrant (the converse follows from Remark 4.5.6.25). Let $\mathscr {E}: \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and let $\mathscr {E}_0 \subseteq \mathscr {E}$ be a subfunctor having the property that, for every integer $n \geq 0$, the inclusion map $\mathscr {E}_0( [n] ) \hookrightarrow \mathscr {E}( [n] )$ is a categorical equivalence. We wish to show that every natural transformation $\alpha _0: \mathscr {E}_0 \rightarrow \mathscr {F}$ can be extended to a map $\alpha : \mathscr {E} \rightarrow \mathscr {F}$.
For each integer $n \geq 0$, let $\mathscr {E}_{n} \subseteq \mathscr {E}$ denote the smallest subfunctor which contains $\mathscr {E}_{0}$ and satisfies $\mathscr {E}_{n}( [m] ) = \mathscr {E}( [m] )$ for $m < n$. Then $\mathscr {E}$ can be written as the union of an increasing sequence of subfunctors
\[ \mathscr {E}_{0} \subseteq \mathscr {E}_{1} \subseteq \mathscr {E}_{2} \subseteq \cdots \]
To complete the proof, it will suffice to show that every natural transformation $\alpha _{n}: \mathscr {E}_{n} \rightarrow \mathscr {F}$ admits an extension $\alpha _{n+1}: \mathscr {E}_{n+1} \rightarrow \mathscr {F}$. Using Proposition 1.1.4.12, we see that the inclusion $\mathscr {E}_{n} \hookrightarrow \mathscr {E}_{n+1}$ is a pushout of the inclusion map
\[ \iota _ n: (\underline{ \mathscr {E}_ n( [n] ) } \times \underline{ \Delta ^ n }) \coprod _{ (\underline{ \mathscr {E}_ n( [n] ) } \times \underline{ \operatorname{\partial \Delta }^ n} )} ( \underline{ \mathscr {E}([n] )} \times \underline{ \operatorname{\partial \Delta }^ n } ) \hookrightarrow \underline{ \mathscr {E}([n] )} \times \underline{ \Delta ^ n}. \]
Consequently, to prove the existence of $\alpha _{n+1}$, we are reduced to solving a lifting problem
\[ \xymatrix { \mathscr {E}_{n}([n]) \ar [d] \ar [r] & \mathscr {F}([n] ) \ar [d]^{\theta _ n} \\ \mathscr {E}( [n] ) \ar [r] \ar@ {-->}[ur] & \varprojlim _{ [m] \hookrightarrow [n] } \mathscr {F}( [m] ). } \]
By virtue of assumption $(\ast _ n)$, it will suffice to show that the inclusion map $\mathscr {E}_{n}([n] ) \hookrightarrow \mathscr {E}( [n] )$ is a categorical equivalence of simplicial sets.
In fact, we will prove a stronger assertion: the inclusion map $\rho _{n}: \mathscr {E}_{n} \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence for every integer $n \geq 0$. Our proof proceeds by induction on $n$, the case $n = 0$ being trivial. To carry out the inductive step, let us assume that $\rho _{n}$ is a levelwise categorical equivalence for some $n \geq 0$; in particular, the inclusion map $\mathscr {E}_{n}( [n] ) \hookrightarrow \mathscr {E}( [n] )$ is a categorical equivalence. Since the collection of categorical equivalences is closed under the formation of coproducts (Corollary 4.5.3.10), it follows that the natural transformation $\iota _{n}$ is a levelwise categorical equivalence. The inclusion map $\mathscr {E}_{n} \hookrightarrow \mathscr {E}_{n+1}$ is a pushout of $\iota _ n$, and is therefore also a levelwise categorical equivalence (Remark 4.5.4.13). Since the collection of (levelwise) categorical equivalences satisfies the two-out-of-three property (Remark 4.5.3.5), it follows that $\rho _{n+1}$ is also a categorical equivalence.
$\square$