Notation 8.6.2.1. Let $\operatorname{\mathcal{D}}$ be a simplicial set equipped with a morphism $\lambda = (\lambda _{-}, \lambda _{+} ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{D}}_{+}$. For every simplicial set $\operatorname{\mathcal{E}}$, we let $\operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ denote the relative exponential of Construction 4.5.9.1. For $n \geq 0$, we will identify $n$-simplices of $\operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ with pairs $(\sigma , f)$, where $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}$ and $f: \Delta ^ n \times _{ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}} } \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism of simplicial sets. Suppose that we are also given a morphism of simplicial sets $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}_{+}$. In this case, we let $\operatorname{Fun}_{ / \operatorname{\mathcal{D}}_{+} }( \operatorname{\mathcal{D}}/ \operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ denote the simplicial subset of $\operatorname{Fun}( \operatorname{\mathcal{D}}/\operatorname{\mathcal{D}}_{-}^{\operatorname{op}}, \operatorname{\mathcal{E}})$ whose $n$-simplices are pairs $(\sigma , f)$ which satisfy the additional condition that the diagram
is commutative.