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!