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.
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