> **Puzzle 131.** ... Again, it may help to draw a graph.

I wanted to see where the overcount in [comment 13](https://forum.azimuthproject.org/discussion/comment/19404/#Comment_19404) came from.

![diagram](https://docs.google.com/drawings/d/e/2PACX-1vSux2Gbr40h292IpzbIXWFHmpqctSTatIMKNvtJi6DHw-vsFKv5tBGjHmfjsJw5EDstfNAZG59zMmR4/pub?w=384&h=239)

- 6:6 of length 0 (the identity transformations)

- 6:6 of length 1 [black] (the transformation arrows on the graph)

- 5:6 of length 2 [blue] (three via F'', three via G) one commuting pair i.e. F'->G->G' = F'->F"->G'

- 2:4 of length 3 [red] (two via F'', two via G) two commuting pairs e.g. F->F'->G->G' = F->F'->F"->G'

- 1:2 of length 4 [green] (one via F'', one via G) these commute

Which is correct?

Are these overcounts?

Or are they distinct natural transformations?

When composing natural transformations do they always commute?

Edit: 'commute' is ambiguous here. It means "produce a commuting square"

or [does composition of natural transformations produce a natural transformation](https://forum.azimuthproject.org/discussion/2249/lecture-45-chapter-3-composing-natural-transformations/p1)?

The answer is, as Robert says, yes!

The details for this are [covered in the subsequent lecture](https://forum.azimuthproject.org/discussion/2249/lecture-45-chapter-3-composing-natural-transformations/p1).

This is different than [commuting of composition](https://forum.azimuthproject.org/discussion/comment/19478/#Comment_19478).

I wanted to see where the overcount in [comment 13](https://forum.azimuthproject.org/discussion/comment/19404/#Comment_19404) came from.

![diagram](https://docs.google.com/drawings/d/e/2PACX-1vSux2Gbr40h292IpzbIXWFHmpqctSTatIMKNvtJi6DHw-vsFKv5tBGjHmfjsJw5EDstfNAZG59zMmR4/pub?w=384&h=239)

- 6:6 of length 0 (the identity transformations)

- 6:6 of length 1 [black] (the transformation arrows on the graph)

- 5:6 of length 2 [blue] (three via F'', three via G) one commuting pair i.e. F'->G->G' = F'->F"->G'

- 2:4 of length 3 [red] (two via F'', two via G) two commuting pairs e.g. F->F'->G->G' = F->F'->F"->G'

- 1:2 of length 4 [green] (one via F'', one via G) these commute

Which is correct?

Are these overcounts?

Or are they distinct natural transformations?

When composing natural transformations do they always commute?

Edit: 'commute' is ambiguous here. It means "produce a commuting square"

or [does composition of natural transformations produce a natural transformation](https://forum.azimuthproject.org/discussion/2249/lecture-45-chapter-3-composing-natural-transformations/p1)?

The answer is, as Robert says, yes!

The details for this are [covered in the subsequent lecture](https://forum.azimuthproject.org/discussion/2249/lecture-45-chapter-3-composing-natural-transformations/p1).

This is different than [commuting of composition](https://forum.azimuthproject.org/discussion/comment/19478/#Comment_19478).