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Notation 5.6.5.16 (The Covariant Transport Representation). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cocartesian fibration of simplicial sets. We let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ denote a covariant transport representation of $U$, regarded as an object of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ (which exists by virtue of Corollary 5.6.5.14). We write $[ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ]$ for the isomorphism class of the diagram $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$, regarded as an object of the set $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$. By virtue of Corollary 5.6.5.15, the isomorphism class $[ \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ]$ is well-defined: that is, it depends only on the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Beware that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is not uniquely determined: in fact, any diagram isomorphic to $\operatorname{Tr}_{\operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}}$ is also a covariant transport representation of $U$ (Remark 5.6.5.11). Nevertheless, it will be convenient to abuse terminology and refer to $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the covariant transport representation of $U$, with the caveat that it is well-defined only up to isomorphism.