One concern also after a wiggly road ahead warning.

Thinking about the example, and about what naturality means with the actual data, I started to think laterally and came to the suspicion that what was sought was [Puzzle 151]( I was thinking, the category of database \\(\mathcal{C}\\)-instances is \\(\mathbf{Set}^\mathbf{2}\\), and the category of \\(\mathcal{D}\\)-instances is \\(\mathbf{Set}^\mathbf{1}\\), i. e. (equivalent to), \\(\mathbf{Set}\\). At the level of instances, forgetting italians would be the functor \\( F: \mathbf{Set}^2 \to \mathbf{Set}\\) defined in the puzzle, and we seek a left adjoint. But I think there's a problem!

What is exactly "\\(\mathbf{2}\\)"? In puzzle 151 I think it is just two isolated identities. But here it is as in [this comment]( So the category of \\(\mathcal{C}\\)-instances is more exactly \\(\mathbf{Set}^{\rightarrow}\\), the [arrow category]( of **Set**, that is, the category of functions and commutative squares. Our functor of precomposing instances with \\(G\\) is (in this example), I believe, the more sophisticated [domain opfibration]( \\(dom: \\mathbf{Set}^{\rightarrow} \to \mathbf{Set}\\), so we are asking for a left adjoint to **that**. Looks hairy.