What is the hom space in over category of simplicial sets?

Question

Let $Set_\Delta$ denote category of simplicial sets, which is enriched in $Set_\Delta$.

Let $B$ be a simplicial set. We can form the over category. $(Set_\Delta)_{/B}$. Then is this also enriched in $Set_\Delta$?

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Answer 1

If $f:X\rightarrow B$ and $g:Y\rightarrow B$ are simplicial sets over $B$ then let $\underline{sSet_B}(X,Y)$ be the sub-simplicial set of $\underline{sSet}(X,Y)$ whose $n$-simplices are the maps $\alpha:X\times\Delta^n\rightarrow Y$ which satisfy $g\circ\alpha=f\circ pr_1$.

Then these sub-simplicial sets are stable under the existing composition in $sSet$, which thus restricts to give you simplicial maps $\circ:\underline{sSet_B}(Y,Z)\times\underline{sSet_B}(X,Y)\rightarrow \underline{sSet_B}(X,Z)$. Moreover, the identity of $X$ in $\underline{sSet}(X,X)$ already lies in $\underline{sSet_B}(X,X)$, and these supply the identities in $sSet_B$. All the diagrams you need to commute do so because they already do in $sSet$.