Variant 5.5.2.3 (Coslices of Simplicial-Categories). Let $\operatorname{\mathcal{C}}$ be a simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$. We define a simplicial category $\operatorname{\mathcal{C}}_{X/}$ as follows:
The objects of $\operatorname{\mathcal{C}}_{X/}$ are pairs $(C,f)$, where $C$ is an object of $\operatorname{\mathcal{C}}$ and $f$ is a vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,C)_{\bullet }$.
Let $(C,f)$ and $(D,g)$ be objects of $\operatorname{\mathcal{C}}_{X/}$. We let $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}( (C,f), (D,g) )_{\bullet }$ denote the simplicial set given by the fiber product
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,D)_{\bullet } } \{ g \} , \]which we regard as a simplicial subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet }$.
Let $(C,f)$, $(D,g)$, and $(E,h)$ be objects of $\operatorname{\mathcal{C}}_{X/}$. Then the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/X}}( (D,g) , (E,h) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}( (C,f) , (D,g) )_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_{X/}}( (C,f) , (E,h) )_{\bullet } \]for the simplicial category $\operatorname{\mathcal{C}}_{X/}$ is given by the restriction of the composition law
\[ \circ : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(D,E)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, E)_{\bullet } \]for the simplicial category $\operatorname{\mathcal{C}}$.