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

**Puzzle 126.** Define \\(\alpha_{\textrm{People}}\\) to be the inclusion map between the sets \\(\\{\textrm{Alice}, \textrm{Bob}, \textrm{Stan}, \textrm{Tyler}\\}\\) and \\(\\{\textrm{Alice}, \textrm{Bob}, \textrm{Stan}, \textrm{Tyler}, \textrm{Mei-Chu}\\}\\). As a restriction of the identity function, all squares commute.

Practically speaking, the inclusion map allows us to extend a database in a consistent way. As long as all relations in the source database are preserved by the target database, we can add new rows.

**Puzzle 127.** Let \\(\alpha_{\textrm{People}}\\) be the identity map on everyone except Stan, who instead gets mapped to Bob. It is notable that the database instance \\(H\\) replaces Stan with Bob as Tyler's friend, which ensures that all induced squares commute.

This seems to be a kind of reduction or compaction, where equivalent rows are coalesced and existing relationships involving the coalesced rows are updated. It reminds me a lot of [minimization of deterministic finite-state automata](https://en.wikipedia.org/wiki/DFA_minimization).

**Puzzle 128.** We can leave everything the same (identity) or project Tyler onto Alice. We cannot project Alice onto Tyler or swap Tyler and Alice, since \\(\mathrm{FriendOf}(\alpha(\textrm{Bob})) = \textrm{Alice} \neq \textrm{Tyler} = \alpha(\mathrm{FriendOf}(\textrm{Bob}))\\).

**Puzzle 126.** Define \\(\alpha_{\textrm{People}}\\) to be the inclusion map between the sets \\(\\{\textrm{Alice}, \textrm{Bob}, \textrm{Stan}, \textrm{Tyler}\\}\\) and \\(\\{\textrm{Alice}, \textrm{Bob}, \textrm{Stan}, \textrm{Tyler}, \textrm{Mei-Chu}\\}\\). As a restriction of the identity function, all squares commute.

Practically speaking, the inclusion map allows us to extend a database in a consistent way. As long as all relations in the source database are preserved by the target database, we can add new rows.

**Puzzle 127.** Let \\(\alpha_{\textrm{People}}\\) be the identity map on everyone except Stan, who instead gets mapped to Bob. It is notable that the database instance \\(H\\) replaces Stan with Bob as Tyler's friend, which ensures that all induced squares commute.

This seems to be a kind of reduction or compaction, where equivalent rows are coalesced and existing relationships involving the coalesced rows are updated. It reminds me a lot of [minimization of deterministic finite-state automata](https://en.wikipedia.org/wiki/DFA_minimization).

**Puzzle 128.** We can leave everything the same (identity) or project Tyler onto Alice. We cannot project Alice onto Tyler or swap Tyler and Alice, since \\(\mathrm{FriendOf}(\alpha(\textrm{Bob})) = \textrm{Alice} \neq \textrm{Tyler} = \alpha(\mathrm{FriendOf}(\textrm{Bob}))\\).