Tag 01JA of the Stacks Project is about gluing schemes. There, a gluing data is defined as a collection of schemes, and isomorphisms between open subsets of those schemes satisfying the cocycle relation. Such a gluing data can be glued to a scheme which I will denote $glue(X)$
The description for the set of morphisms from $glue(X)$ to $Y$ seems like some kind of adjoint functor/universal property. However without a notion of morphism of gluing data, we don't have a category of gluing data.
My question is: is there a way to define morphisms of gluing data so that $glue(X)$ is a functor that has an adjoint?