Kerodon

Notation 5.2.2.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets and let $f: C \rightarrow D$ be an edge of the simplicial set $\operatorname{\mathcal{C}}$. Applying Proposition 5.2.2.8, we conclude that the collection of functors $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ which are given by covariant transport along $f$ comprise a single isomorphism class in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )$. We will denote this isomorphism class by $[f_{!}]$, which we regard as an element of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{E}}_{C}, \operatorname{\mathcal{E}}_{D} )^{\simeq })$. We will often use the notation $f_{!}$ to denote a particular choice of representative of this isomorphism class: that is, a particular choice of functor $\operatorname{\mathcal{E}}_ C \rightarrow \operatorname{\mathcal{E}}_{D}$ which is given by covariant transport along $f$.