Example 8.6.2.13 (05JH): Functoriality

Example 8.6.2.13 (Functoriality). In the situation of Corollary 8.6.2.12, suppose we are given a commutative diagram

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

where $U$ and $V$ are cocartesian fibrations, where $F$ carries $U$-cocartesian morphisms of $\operatorname{\mathcal{D}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{E}}$. Then there an essentially unique functor $F^{\dagger }: \operatorname{\mathcal{D}}^{\dagger } \rightarrow \operatorname{\mathcal{E}}^{\dagger }$ satisfying $V^{\dagger } \circ F^{\dagger } = U^{\dagger }$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{D}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T_0} \ar [d]^{F^{\dagger } \times \operatorname{id}} & \operatorname{\mathcal{D}}\ar [d]^{F} \\ \operatorname{\mathcal{E}}^{\dagger } \times _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [r]^-{T} & \operatorname{\mathcal{E}}} \]

commutes up to isomorphism (over $\operatorname{\mathcal{C}}$). Moreover, $F^{\dagger }$ automatically carries $U^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{D}}^{\dagger }$ to $V^{\dagger }$-cartesian morphisms of $\operatorname{\mathcal{E}}^{\dagger }$.