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Proposition Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a fibration in sets and let $\chi _{U}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ be the contravariant transport functor of Construction Then there is a unique functor $F: \operatorname{\mathcal{D}}\rightarrow \int ^{\operatorname{\mathcal{C}}} \chi _{U}$ with the following properties:


The diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}\ar [rr]^{F} \ar [dr]^{U} & & \int ^{\operatorname{\mathcal{C}}} \chi _{U} \ar [dl] \\ & \operatorname{\mathcal{C}}& } \]

is strictly commutative (where the right vertical map is the forgetful functor).


For each object $X \in \operatorname{\mathcal{C}}$, the induced map

\[ \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} ) \xrightarrow {F} \operatorname{Ob}( \{ X\} \times _{ \operatorname{\mathcal{C}}} ( \int ^{\operatorname{\mathcal{C}}} \chi _ U ) ) = \chi _ U(X) \]

is equal to the identity.

Moreover, the functor $F$ is an isomorphism of categories.

Proof. Note that the functor $F$ is automatically unique if it exists: its value on each object of $\operatorname{\mathcal{D}}$ is determined by condition $(b)$, and its restriction to $\operatorname{Hom}_{\operatorname{\mathcal{D}}}( \widetilde{X}, \widetilde{Y} )$ is determined by condition $(a)$ (since the forgetful functor $\int ^{\operatorname{\mathcal{C}}} \chi _ U \rightarrow \operatorname{\mathcal{C}}$ is faithful). To prove existence, it suffices to observe that if $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ is a morphism in $\operatorname{\mathcal{D}}$ having image $f: X \rightarrow Y$ in the category $\operatorname{\mathcal{C}}$, then the function $f^{\ast }: \operatorname{Ob}( \operatorname{\mathcal{D}}_{Y} ) \rightarrow \operatorname{Ob}(\operatorname{\mathcal{D}}_{X} )$ carries $\widetilde{Y}$ to $\widetilde{X}$ (which is immediate from the definition of $f^{\ast }$). We will complete the proof by showing that $F$ is an isomorphism of categories. It follows from $(b)$ that $F$ is bijective at the level of objects. It will therefore suffice to show that, for every pair of objects $\widetilde{X}, \widetilde{Y} \in \operatorname{\mathcal{D}}$, the induced map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \widetilde{X}, \widetilde{Y} ) \rightarrow \operatorname{Hom}_{ \int ^{\operatorname{\mathcal{C}}} \chi _ U }( F(\widetilde{X}), F( \widetilde{Y} ) ) \]

is a bijection. Setting $X = U( \widetilde{X} )$ and $Y = U( \widetilde{Y} )$, we can identify the target of $\theta $ with the subset of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ consisting of those morphism $f$ which satisfy $f^{\ast }( \widetilde{Y} ) = \widetilde{X}$. The bijectivity of $\theta $ now follows from Remark $\square$