Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

Notation 5.5.2.5. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a locally cartesian fibration of simplicial sets and let $e: X \rightarrow Y$ be an edge of the simplicial set $\operatorname{\mathcal{D}}$. It follows from Proposition 5.5.2.3 that the collection of functors $\operatorname{\mathcal{C}}_{Y} \rightarrow \operatorname{\mathcal{C}}_{X}$ which are given by contravariant transport along $e$ comprise a single isomorphism class in the $\infty$-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{Y}, \operatorname{\mathcal{C}}_{X} )$. We will denote this isomorphism class by $[e^{\ast }]$, which we regard as an element of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}_{Y}, \operatorname{\mathcal{C}}_{X} )^{\simeq } )$. We will often use the notation $e^{\ast }$ to denote a particular choice of representative of this isomorphism class: that is, a particular choice of functor $\operatorname{\mathcal{C}}_ Y \rightarrow \operatorname{\mathcal{C}}_{X}$ which is given by contravariant transport along $e$.