Still trying to figure out Puzzles 150 - 155 (note that the numbering of the last two puzzles is repeated in the original post, I am extending it here). To wrap up the solutions so far:

**Puzzle 150:**

> \\( R(S) = (S, 1) \\) for any set \\( S \\) and \\( R(f) = (f, 1_1) \\) for any function \\(f: S \to T\\), where \\(1_1\\) means the identity function from our one-element set to itself.

By the way, it seems to me that this identity differs from Dan's solution, where

> \\(!\\) denotes the unique function from any set to the singleton set \\(\\{\bullet\\}\\).

**Puzzle 151:** ???

**Puzzle 152:** Does not exist?

**Puzzle 153:** The diagonal functor \\( \Delta \\).

**Puzzle 154:** Also the diagonal functor \\( \Delta \\).

**Puzzle 155:** ???

I think that these examples are really great because they are simple and concrete. But I still find it very hard to start reasoning about adjunctions, I can only think: "what would be a generous/selfish way to come back?". And once I have a candidate, it is hard to start a proving that it is an adjoint. Is there any reference where we can find complete proofs of all this?

**Puzzle 150:**

> \\( R(S) = (S, 1) \\) for any set \\( S \\) and \\( R(f) = (f, 1_1) \\) for any function \\(f: S \to T\\), where \\(1_1\\) means the identity function from our one-element set to itself.

By the way, it seems to me that this identity differs from Dan's solution, where

> \\(!\\) denotes the unique function from any set to the singleton set \\(\\{\bullet\\}\\).

**Puzzle 151:** ???

**Puzzle 152:** Does not exist?

**Puzzle 153:** The diagonal functor \\( \Delta \\).

**Puzzle 154:** Also the diagonal functor \\( \Delta \\).

**Puzzle 155:** ???

I think that these examples are really great because they are simple and concrete. But I still find it very hard to start reasoning about adjunctions, I can only think: "what would be a generous/selfish way to come back?". And once I have a candidate, it is hard to start a proving that it is an adjoint. Is there any reference where we can find complete proofs of all this?