Kerodon

Notation 5.2.2.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$. It follows from Proposition 5.2.2.17 that the collection of functors $\operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ which are given by contravariant transport along $f$ comprise a single isomorphism class in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )$. We will denote this isomorphism class by $[f^{\ast }]$, which we regard as an element of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{C} )^{\simeq } )$. We will often use the notation $f^{\ast }$ to denote a particular choice of representative of this isomorphism class: that is, a particular choice of functor $\operatorname{\mathcal{E}}_ D \rightarrow \operatorname{\mathcal{E}}_{C}$ which is given by contravariant transport along $f$.