# Kerodon

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Variant 9.10.6.8. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an opfibration in sets. We define a functor $\chi _{U}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ as follows:

• For each object $X \in \operatorname{\mathcal{C}}$, we define $\chi _{U}(X)$ to be the set of objects $\operatorname{Ob}( \operatorname{\mathcal{D}}_{X} )$.

• For each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we define $\chi _{U}(f)$ to be the function $f_!: \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} ) \rightarrow \operatorname{Ob}( \operatorname{\mathcal{D}}_{Y} )$ of Variant 9.10.6.2.

We will refer to $\chi _{U}$ as the transport representation associated to $U$.