>**Puzzle 149.** Again suppose \\(F: \mathbf{Set} \to \mathbf{1}\\) is the functor that sends every set to \\(\star\\) and every function to \\(1_\star\\). A _left_ adjoint \\(L : \mathbf{1} \to \mathbf{Set} \\) is a functor for which there's a natural one-to-one correspondence between functions

>\[ m: L(\star) \to S \]

>and morphisms

>\[ n: \star \to F(S) \]

>for every set \\(S\\). On the basis of this, try to figure out all the left adjoints of \\(F\\).

Demanding that \\(L(\star)\to S\\), for all \\(S\\) is the same thing as asking for a set \\(T\\) such that, for all \\(S\\), \\(T \to S\\) holds.

The only such set is the empty set, so the left adjoint to \\(F: \mathbf{Set} \to \mathbf{1}\\) is a functor taking \\(\mathbf{1}\\) to the empty set.

\\[

L(\star) = \varnothing, \\\\

L(1\_\star) = 1\_\varnothing

\\]

>\[ m: L(\star) \to S \]

>and morphisms

>\[ n: \star \to F(S) \]

>for every set \\(S\\). On the basis of this, try to figure out all the left adjoints of \\(F\\).

Demanding that \\(L(\star)\to S\\), for all \\(S\\) is the same thing as asking for a set \\(T\\) such that, for all \\(S\\), \\(T \to S\\) holds.

The only such set is the empty set, so the left adjoint to \\(F: \mathbf{Set} \to \mathbf{1}\\) is a functor taking \\(\mathbf{1}\\) to the empty set.

\\[

L(\star) = \varnothing, \\\\

L(1\_\star) = 1\_\varnothing

\\]