Naturality and definability III

We explore the relationship between the notions of naturality from category theory and definability from model theory. We study their interactions and present three main results. First, we show that under some mild conditions, naturality implies definability. Second, using reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we demonstrate that if $F$ is a natural construction on a class $\mathcal K$ of structures, represented by some formula, then it is uniformly definable without the need for extra parameters. Our results resolve some questions posed by Hodges and Shelah.