Jonathan wrote:

> I was always quite confused by natural transformations, but it seems they're just a selection of morphisms in the target category? How interesting!

Yes!

Well, a **transformation** between functors \\(F, G : \mathcal{C} \to \mathcal{D}\\) is just an arbitrary selection of a morphism \\(\alpha_x : F(x) \to G(x) \\) in the target category for each object \\(x\\) in the source category. That's enough to create the vertical 'struts' in this picture:

But it's **natural** when the squares with these struts as their vertical sides _commute_. And that's the cool part.

Your answer to Puzzle 126 is very nice, and the nugget of insight lurks here:

> Practically speaking, the inclusion map allows us to extend a database in a consistent way.

The commuting of the squares, which makes our transformation 'natural', is a kind of _consistency condition!_

Your answer to Puzzle 127 is also very nice:

> This seems to be a kind of reduction or compaction, where equivalent rows are coalesced and existing relationships involving the coalesced rows are updated.

Right, and the commuting of the naturality squares says we are doing this in a _consistent_ way.

> It is notable that the database instance \\(H\\) replaces Stan with Bob as Tyler's friend, which ensures that all induced squares commute.

Maybe someone can expound a bit more on the notion of 'consistency' that I'm vaguely alluding to. All I mean is that a bunch of squares commute... but it's good to try to tease out the meaning of this, and express it in something resembling plain English.

> I was always quite confused by natural transformations, but it seems they're just a selection of morphisms in the target category? How interesting!

Yes!

Well, a **transformation** between functors \\(F, G : \mathcal{C} \to \mathcal{D}\\) is just an arbitrary selection of a morphism \\(\alpha_x : F(x) \to G(x) \\) in the target category for each object \\(x\\) in the source category. That's enough to create the vertical 'struts' in this picture:

But it's **natural** when the squares with these struts as their vertical sides _commute_. And that's the cool part.

Your answer to Puzzle 126 is very nice, and the nugget of insight lurks here:

> Practically speaking, the inclusion map allows us to extend a database in a consistent way.

The commuting of the squares, which makes our transformation 'natural', is a kind of _consistency condition!_

Your answer to Puzzle 127 is also very nice:

> This seems to be a kind of reduction or compaction, where equivalent rows are coalesced and existing relationships involving the coalesced rows are updated.

Right, and the commuting of the naturality squares says we are doing this in a _consistent_ way.

> It is notable that the database instance \\(H\\) replaces Stan with Bob as Tyler's friend, which ensures that all induced squares commute.

Maybe someone can expound a bit more on the notion of 'consistency' that I'm vaguely alluding to. All I mean is that a bunch of squares commute... but it's good to try to tease out the meaning of this, and express it in something resembling plain English.