>**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 \$