its simplicial set of objects is constant because $U$ and $F$ took note of the identity loops.

In more detail, let $ob : Cat \to Sets$ be the functor that picks out the set of objects of a small category, then $ob(S(\mathbb{A})) :  \mathbf{\Delta}^{op} \to Sets$ is a constant functor with value the set $ob(\mathbb{A})$ of objects of $\mathbb{A}$. More exactly it is a discrete simplicial set, since all its face and degeneracy maps are bijections. Using those bijections to identify the possible different sets of objects, yields a constant simplicial set where all the face and degeneracy maps are identity maps, i.e. we do have a *constant* simplicial set.

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Let $\mathcal{B} : \mathbf{\Delta}^{op} \to Cat$ be a simplicial object in $Cat$ such that $ob(\mathcal{B})$ is a constant simplicial set with value $B_0$, say. For each pair $(x,y)\in B_0$, let $$\mathcal{B}(x,y)_{n} = \{\sigma \in \mathcal{B}_{n} | ~  {\rm dom}(\sigma) = x, {\rm codom}(\sigma) = y\},$$ where, of course, ${\rm dom}$ refers to the domain function in $\mathcal{B}_n$ since otherwise ${\rm dom}(\sigma)$ would have no meaning, similarly for $\rm codom$.

\(i) The collection $\{\mathcal{B}(x,y)_n |~ n \in \mathbb{N}\}$ has the structure of a simplicial set $\mathcal{B}(x,y)$ with face and degeneracies 