Warning 6.2.4.5. The formulation of Proposition 6.2.4.4 is slightly imprecise, since the contravariant transport functor $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ is only well-defined up to isomorphism. For the proof, it will be enough to assume that there is some contravariant transport functor which carries $\operatorname{\mathcal{E}}'_{D}$ into $\operatorname{\mathcal{E}}'_{C}$. If the subcategory $\operatorname{\mathcal{E}}' \subseteq \operatorname{\mathcal{E}}$ is replete, this implies that any contravariant transport functor $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ has the same property.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$