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For a simplicial set $X$, let $\text{el X}$ be its category of elements, whose objects are pairs $([n], x\in X_n)$. Let $\text{(el X)}_{nd}$ denote the full subcategory comprising objects $([m], y\in X_m)$ where $y$ is nondegenerate. I'd like to prove that the inclusion $i :\text{(el X)}_{nd}\hookrightarrow \text{el X}$ is a final functor by showing $$([n], x)\downarrow i$$ is nonempty and connected.

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This is not true. Consider the simplicial set $X=\Delta^2/\Delta^{\{0,1\}}$. Then there is a degenerate $1$-simplex $x$ corresponding to the collapsed edge of $\Delta^2$, which is both the degeneracy on the initial vertex $a$ of $X$ and the face $d_2$ of the unique nondegenerate $2$-simplex $z$ of $X$. However, there is no way to connect $([1],x)\to([0],a)$ with $d^2\colon([1],x)\to([2],z)$ in $([1],x)\downarrow i$.

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