[Michael wrote](https://forum.azimuthproject.org/discussion/comment/20726/#Comment_20726):

> I have a question that has been plaguing me for awhile and this lecture has brought it back to my attention. It seems like once we start talking about categories, all laws or equations are presented as natural transformations like the associator and unitors.

>

> I don't get why we present it as natural transformations.

>

> It seems like this is something important when learning category but I haven't fully understood while building up from definitions.

I've seen similar things crop up in topology; for instance, a topological space \\(T\\) is Hausdorff iff the diagonal \\(\Delta = \\{(x, x) \mid x \in T\\}\\) is a closed set in \\(T \times T\\). There's something very pleasant about being able to embed a property, a kind of external observation about an object, as an internal feature of that object's structure. Giving laws in terms of natural transformations has the same feel: we're specifying that the category under consideration possesses some extra internal features that reify desirable properties.

Maybe a slightly more accurate comparison is with the idea of a "metric space" vs. a "metrizable space". A metric space is a space coupled with a metric; a metrizable space is merely one that can be generated by some undetermined metric. The latter has the feel of a property; the former has the feel of structure.

(In other news, I'm getting forum notifications again. Huzzah!)

> I have a question that has been plaguing me for awhile and this lecture has brought it back to my attention. It seems like once we start talking about categories, all laws or equations are presented as natural transformations like the associator and unitors.

>

> I don't get why we present it as natural transformations.

>

> It seems like this is something important when learning category but I haven't fully understood while building up from definitions.

I've seen similar things crop up in topology; for instance, a topological space \\(T\\) is Hausdorff iff the diagonal \\(\Delta = \\{(x, x) \mid x \in T\\}\\) is a closed set in \\(T \times T\\). There's something very pleasant about being able to embed a property, a kind of external observation about an object, as an internal feature of that object's structure. Giving laws in terms of natural transformations has the same feel: we're specifying that the category under consideration possesses some extra internal features that reify desirable properties.

Maybe a slightly more accurate comparison is with the idea of a "metric space" vs. a "metrizable space". A metric space is a space coupled with a metric; a metrizable space is merely one that can be generated by some undetermined metric. The latter has the feel of a property; the former has the feel of structure.

(In other news, I'm getting forum notifications again. Huzzah!)