Take the 2-minute tour ×   Let $FU_\bullet:Cat\to Cat_\Delta$ be
the bar construction assigned to the comonad $FU$ determined by
free-forgetful adjunction $F:Quiv\rightleftarrows Cat:U$. The
restriction of $FU_\bullet$ to the full subcategory $\Delta$
(which is isomorphic to the category of finite nonempty
ordinals) naturally determines a colimit-preserving functor
$\mathfrak{C}:Set_\Delta=Set^{\Delta^{op}}\to Cat_\Delta$. The
right adjoint of this functor is called $\mathfrak{N}$, the
homotopy-coherent nerve.  Identify $Cat$ (not by $FU_\bullet$)
with the full subcategory of $Cat_\Delta$ spanned by those
simplicially enriched categories with discrete hom-spaces.
Also, recall the definition of the right cone $X^\triangleright$
on a simplicial set $X$ is the join $X\star \Delta^0$. This
determines an obvious natural map $X\to X^\triangleright$.  Let
$X\to \Delta^1=\mathfrak{N}([1])$ be an object of
$(Set_\Delta\downarrow \Delta^1)$, and let $\epsilon:\mathfrak{C
}(\Delta^1)=\mathfrak{C}(\mathfrak{N}([1])\to [1