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](https://forum.azimuthproject.org/discussion/2264/lecture-48-chapter-3-adjoint-functors). 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](https://forum.azimuthproject.org/discussion/comment/19672/#Comment_19672). So the category of \$$\mathcal{C}\$$-instances is more exactly \$$\mathbf{Set}^{\rightarrow}\$$, the [arrow category](https://ncatlab.org/nlab/show/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](https://ncatlab.org/nlab/show/domain+opfibration) \$$dom: \\mathbf{Set}^{\rightarrow} \to \mathbf{Set}\$$, so we are asking for a left adjoint to **that**. Looks hairy.