Notation 4.6.8.1. Let $K$ be a simplicial set. We define a simplicial category $\operatorname{\mathcal{E}}[K]$ as follows:
The category $\operatorname{\mathcal{E}}[K]$ has exactly two objects, which we will denote by $x$ and $y$.
The morphism spaces in $\operatorname{\mathcal{E}}[K]$ are given by the formulae
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}[K]}( x, x)_{\bullet } = \{ \operatorname{id}_{x} \} \quad \quad \operatorname{Hom}_{K}(y,y)_{\bullet } = \{ \operatorname{id}_ y \} \]\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}[K]}(x,y)_{\bullet } = K \quad \quad \operatorname{Hom}_{K}( y,x)_{\bullet } = \emptyset . \]