The Stacks project

Remark 27.2.3. There is a functoriality property for the constructions explained in Lemmas 27.2.1 and 27.2.2. Namely, suppose given two collections of data $(f_ U : X_ U \to U, \rho ^ U_ V)$ and $(g_ U : Y_ U \to U, \sigma ^ U_ V)$ as in Lemma 27.2.1. Suppose for every $U \in \mathcal{B}$ given a morphism $h_ U : X_ U \to Y_ U$ over $U$ compatible with the restrictions $\rho ^ U_ V$ and $\sigma ^ U_ V$. Functoriality means that this gives rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_ U$, where $f : X \to S$ is obtained from the datum $(f_ U : X_ U \to U, \rho ^ U_ V)$ and $g : Y \to S$ is obtained from the datum $(g_ U : Y_ U \to U, \sigma ^ U_ V)$.

Similarly, suppose given two collections of data $(f_ U : X_ U \to U, \mathcal{F}_ U, \rho ^ U_ V, \theta ^ U_ V)$ and $(g_ U : Y_ U \to U, \mathcal{G}_ U, \sigma ^ U_ V, \eta ^ U_ V)$ as in Lemma 27.2.2. Suppose for every $U \in \mathcal{B}$ given a morphism $h_ U : X_ U \to Y_ U$ over $U$ compatible with the restrictions $\rho ^ U_ V$ and $\sigma ^ U_ V$, and a morphism $\tau _ U : h_ U^*\mathcal{G}_ U \to \mathcal{F}_ U$ compatible with the maps $\theta ^ U_ V$ and $\eta ^ U_ V$. Functoriality means that these give rise to a morphism of schemes $h : X \to Y$ over $S$ restricting back to the morphisms $h_ U$, and a morphism $h^*\mathcal{G} \to \mathcal{F}$ restricting back to the maps $h_ U$ where $(f : X \to S, \mathcal{F})$ is obtained from the datum $(f_ U : X_ U \to U, \mathcal{F}_ U, \rho ^ U_ V, \theta ^ U_ V)$ and where $(g : Y \to S, \mathcal{G})$ is obtained from the datum $(g_ U : Y_ U \to U, \mathcal{G}_ U, \sigma ^ U_ V, \eta ^ U_ V)$.

We omit the verifications and we omit a suitable formulation of “equivalence of categories” between relative glueing data and relative objects.