@ThomasRead

Yes, thanks for pointing this out. Continuing with **Puzzle 18**, if \\(f_{\ast}\\) has a left-adjoint \\(g_{\ast}\\), then \\(f\\) must be surjective.

From before, we know that if \\(f_{\ast}\\) has a left-adjoint \\(g_{\ast}\\), then \\(f\\) must be injective. Together, we have that \\(f\\) must be bijective.

Conversely, if \\(f\\) is bijective, so is \\(f_{\ast}\\), and thus has its inverse for a left-adjoint.

Therefore, the functions \\(f\\) for which \\(f_{\ast}\\) has a left-adjoint are exactly those that are bijective.

Yes, thanks for pointing this out. Continuing with **Puzzle 18**, if \\(f_{\ast}\\) has a left-adjoint \\(g_{\ast}\\), then \\(f\\) must be surjective.

From before, we know that if \\(f_{\ast}\\) has a left-adjoint \\(g_{\ast}\\), then \\(f\\) must be injective. Together, we have that \\(f\\) must be bijective.

Conversely, if \\(f\\) is bijective, so is \\(f_{\ast}\\), and thus has its inverse for a left-adjoint.

Therefore, the functions \\(f\\) for which \\(f_{\ast}\\) has a left-adjoint are exactly those that are bijective.