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I do not understand the proof of Variant 4.2.3.16 of Higher Topos Theory by Jacob Lurie, and I need help.

Variant 4.2.3.16 asserts the following:

($\diamond$) Let $K$ be a finite simplicial set. There is a cofinal map $N(A)\to K$, where $A$ is a finite poset.

The proof proceeds as follows:

(1). Consider the following property for a simplicial set $K$:

($\ast$) Every nondegenerate simplex $\Delta^n\to K$ is a monomorphism.

If K satisfies ($\ast$), then $K$ satisfies the conclusion of ($\diamond$).

(2). We show that, for each finite simplicial set $K$, there is a cofinal map $\widetilde{K}\to K$, where $\widetilde{K}$ is finite and satisfies ($\ast$).

I am fine with step (1). But I don't understand the proof of (2).

Let me explain Lurie's proof of (2). He argues that we can prove (2) by induction on the number of nondegenerate simplices of $K$. According to him, this is because if $K$ can be written as $K=K_0\amalg _{\partial\Delta^n} \Delta^n$, where $K_0$ satisfies ($\ast$), then $K$ also satisfies ($\ast$). To prove this, he choose a cofinal map $\widetilde{K_0}\to K_0$, where $\widetilde{K}_0$ is a finite simplicial set satisfying ($\ast$), and claims that the map $$\widetilde{K}=(\widetilde{K}_0\times \Delta^n)\amalg _{\partial \Delta^n} \Delta^n\to K$$ witnesses property ($\ast$).

The problem is, I do not understand what the map $\partial \Delta^n\to \widetilde{K}_0$ used in the definition of $\widetilde{K}$ is. Can someone explain what this map is? (Or does anyone know how to prove (2) or ($\diamond$) in different ways?)

Thanks in advance.

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    $\begingroup$ I think that you can take $A$ to be the coend of the functor $\mathbf{\Delta}^{\text{op}}\times\mathbf{\Delta}\to\text{POSet}$ sending $(n,m)$ to $X_n\times\Xi_m$, where $\Xi_m$ is the poset of nonempty chains of nonempty subsets of $[m]$. However, it takes some work to verify this. $\endgroup$
    – Neil Strickland
    Commented Jul 2, 2023 at 9:33
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    $\begingroup$ As a small point, Kerodon requires $\tilde{K} \to K$ to be a trivial fibration, but the construction given won't produce a finite simplicial set if $K$ is finite. However, Kerodon does have a completely different construction which shows that a finite simplicial set is a localization of a finite poset, which gives the same result. $\endgroup$
    – daniel gratzer
    Commented Jul 2, 2023 at 10:04
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    $\begingroup$ @danielgratzer Thanks! I was not aware of the stronger result in Kerodon. If you make your comment into an answer, I will gladly accept it. $\endgroup$
    – Ken
    Commented Jul 3, 2023 at 2:14

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[Rephrasing my comment as an answer]

While I cannot speak for the actual proof, two points are worth noting. First, a similar construction appears in Kerodon (https://kerodon.net/tag/02QA) but there (1) $\widetilde{K} \to K$ is required to be a trivial fibration but (2) it doesn't restrict correctly to when $K$ is finite.

The second point is that Kerodon also includes a stronger proof of this same fact (https://kerodon.net/tag/02MU). The construction is quite different but should suffice as a replacement.

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