Owen wrote:

> In the discussion after [Lecture 38](https://forum.azimuthproject.org/discussion/comment/18986/#Comment_18986), we were looking at some slides from a talk by David Spivak about databases, and he mentioned the Grothendieck construction that converts an instance \\(I:\mathcal{D}\to \mathbf{Set}\\) to another category \\(\mathrm{Gr}(I)\\) equipped with a functor \\(\mathrm{Gr}(I)\to \mathcal{D}\\). It would be easy to compose this with the functor \\(\mathcal{D}\to\mathcal{C}\\) and get at least what he calls a "semi-instance" \\(\mathrm{Gr}(I) \to \mathcal{C}\\). Does this semi-instance correspond to the actual instance \\(\mathcal{C}\to \mathbf{Set}\\) we'll get with Kan extensions later?

I don't think so. A functor \\(\mathrm{Gr}(I) \to \mathcal{C}\\) is pretty darn different from a functor \\(\mathcal{C}\to \mathbf{Set}\\) .

An object of \\(\mathrm{Gr}(I)\\) is an object \\(d\\) of \\(\mathcal{D}\\) together with an element of the set \\(I(d)\\). So, a functor \\(\mathrm{Gr}(I) \to \mathcal{C}\\) takes an object \\(d\\) of \\(\mathcal{D}\\) together with an element of \\(I(d)\\) and spits out an object in \\(\mathcal{C}\\). On the other hand, a functor \\(\mathcal{C}\to \mathbf{Set}\\) takes an object of \\(\mathcal{C}\\) and spits out a set. These are pretty darn different things to do!