Proposition 5.5.2.16. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial categories. Let $X_0$ and $Y_0$ denote the cone points of $\operatorname{\mathcal{D}}^{\triangleleft }$ and $\operatorname{\mathcal{D}}^{\triangleright }$, respectively. Then:
For every object $Y \in \operatorname{\mathcal{C}}$, postcomposition with the right cone contraction functor $V: (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ of Construction 5.5.2.15 induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$\} } \ar [d]^-{\sim } \\ \{ \textnormal{Simplicial functors $G: \operatorname{\mathcal{D}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ with $G(Y_0)=Y$} \} } \]
For every object $X \in \operatorname{\mathcal{C}}$, postcomposition with the left cone contraction functor $V': (\operatorname{\mathcal{C}}_{X/})^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of Construction 5.5.2.15 induces a bijection
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{X/}$\} } \ar [d]^-{\sim } \\ \{ \textnormal{Simplicial functors $G: \operatorname{\mathcal{D}}^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $G(X_0)=X$} \} } \]
Proof.
We will prove the first assertion; the proof of the second is similar. Fix a simplicial functor $G: \operatorname{\mathcal{D}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ and set $Y = G(Y_0)$. We wish to show that there is a unique simplicial functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}_{/Y}$ for which the composition
\[ \operatorname{\mathcal{D}}^{\triangleright } \xrightarrow { F^{\triangleright } } ( \operatorname{\mathcal{C}}_{/Y} )^{\triangleright } \xrightarrow {V} \operatorname{\mathcal{C}} \]
is equal to $G$. For each object $D \in \operatorname{\mathcal{D}}$, the simplicial functor $G$ induces a morphism of simplicial sets
\[ \Delta ^{0} = \operatorname{Hom}_{ \operatorname{\mathcal{D}}^{\triangleright } }( D, Y_0)_{\bullet } \xrightarrow {G} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), G(Y_0) )_{\bullet }, \]
which we can identify with a vertex $f$ of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), Y)_{\bullet }$. The simplicial functor $F$ is then given on objects by the formula $F(D) = (G(D), f)$, and is determined on morphisms by the requirement that the composition
\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( D, E)_{\bullet } \xrightarrow {F} \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/Y} }( F(D), F(E) )_{\bullet } \subseteq \operatorname{Hom}_{\operatorname{\mathcal{C}}}( G(D), G(E) )_{\bullet } \]
coincides with the map of simplicial sets determined by the simplicial functor $G$.
$\square$