Kerodon

Remark 7.4.3.7. In the statement of Theorem 7.4.3.6, the covariant refraction diagram $F: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} }$ and the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{QC}}$ are only well-defined up to isomorphism (as objects of the $\infty $-categories $\operatorname{Fun}( \operatorname{\mathcal{E}}, \overline{\operatorname{\mathcal{E}}}_{ {\bf 1} })$ and $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\triangleright }, \operatorname{\mathcal{QC}})$, respectively). However, conditions $(1)$ and $(2)$ depend only on their isomorphism classes (see Exercise 6.3.1.11 and Corollary 7.1.2.14).