composition on simplicially enriched categories.

Question

Let $\mathcal{C}$ be a simplicially enriched category i.e a Functor: $\Delta^{op}\to Cat$ with constant objects. Let $x,y \in ob(\mathcal{C}[n])$. We want to define a simplicial set $Hom_{\mathcal{C}}(x,y)$ and a composition on it. The Definition of the simplicial set is easy, just define $Hom_{\mathcal{C}}(x,y)_n = Hom_{\mathcal{C}[n]}(x,y)$. But now I need to define a composition i.e a map (s.t on simplicies it is just composition in $\mathcal{C}[n]$)

$\circ Hom_{\mathcal{C}}(x,y) \times Hom_{\mathcal{C}}(y,z) \to Hom_{\mathcal{C}}(x,z$).

Here I have no idea how to do this. Not only this, I dont even understand that statement. $Hom_{\mathcal{C}}(x,y)$ is a simplicial set, not a set, hence it is a Functor $\Delta^{op}\to Set$, not a set, hence it has no element. But a Map like $\circ$ is defined between sets, not between Functors. Anyone knows what is meant here, and how to do that, I am a bit confused about this

Answer 1

A simplicial set $X$ is a functor $X\colon\Delta^{op}\rightarrow\mathbf{Set}$, i.e. it consists of sets $X_n$ for all $n\ge0$ and maps $\sigma^{\ast}\colon X_n\rightarrow X_m$ for any order-preserving map $\sigma\colon[m]\rightarrow[n]$ in a functorial manner. For simplicial sets $X$ and $Y$, a simplicial map $f\colon X\rightarrow Y$ is a natural transformation of functors $\Delta^{op}\rightarrow\mathbf{Set}$, i.e. it consists of maps $f_n\colon X_n\rightarrow Y_n$ for all $n\ge0$ s.t. $f_m\sigma^{\ast}=\sigma^{\ast}f_n$ for any order-preserving map $\sigma\colon[m]\rightarrow[n]$. If this causes confusion, I recommend reviewing simplicial sets again before worrying about simplicial categories.

The product of simplicial sets is defined object-wise, so we are looking for maps $\mathrm{Hom}_{\mathcal{C}[n]}(x,y)\times\mathrm{Hom}_{\mathcal{C}[n]}(y,z)\rightarrow\mathrm{Hom}_{\mathcal{C}[n]}(x,z)$ for all $n\ge0$, which we take to be the composition maps in the category $\mathcal{C}[n]$. The naturality condition is satisfied for an order-preserving map $\sigma\colon[m]\rightarrow[n]$ induces a functor $\mathcal{C}[n]\rightarrow\mathcal{C}[m]$ and functors preserve composition.