Sophie wrote:

> In the example of transforming databases from [Lecture 41](https://forum.azimuthproject.org/discussion/2230/lecture-41-chapter-3-composing-functors), the functor that ignores departmental data is full and faithful but not essentially surjective. So it "forgets at most properties". I can't figure out how to think of departmental data as a property but maybe I am trying to take the analogy too far?

You can never take this analogy - or more precisely, this concept of 'stuff, structure and properties' - too far. But I'm a bit confused.

Let me remember what we were talking about. We created a category \\(\mathcal{C}\\) by taking the free category on this graph:



and a category \\(\mathcal{D}\\) by taking the free category on this graph:



We got an obvious 'inclusion' functor \\(G : \mathcal{D} \to \mathcal{C}\\).

It sounds like you're asking whether \\(G\\) is full, faithful and/or essentially surjective, and asking what this means in terms of stuff, structure and properties. That's fun to think about.

But here's another fun thing to think about! "Composing with \\(G\\)" turns databases built on the schema \\(\mathcal{C}\\) into databases built on the schema \\(\mathcal{D}\\). In other words, we have a functor

\[ \textrm{ composing with } G \; : \mathbf{Set}^\mathcal{C} \to \mathbf{Set}^\mathcal{D} \]

We can also ask if _this_ functor is full, faithful and/or essentially surjective, and what _this_ means in terms of stuff, structure and properties.

I think I was confused, at first, about what you were thinking about. Now I think it's really the first, since \\(G\\) is indeed full, faithful but not essentially surjective.

I have more to say, but not now, since it's Brandon Coya's graduation and I have to go put a hood on him!