[John Baez](https://forum.azimuthproject.org/profile/17/John%20Baez) wrote:

> I believe that if the formulas give well-defined functions, these are necessarily the desired adjoints!

In puzzle 18, we have seen that \\(f_!\\) doesn't have a left adjoint if \\(f\\) is not injective (comments [5](https://forum.azimuthproject.org/discussion/comment/16501/#Comment_16501) and [6](https://forum.azimuthproject.org/discussion/comment/16509/#Comment_16509)).

Does this mean that the function given by the formula is not well-defined?

[Marc Kaufmann](https://forum.azimuthproject.org/profile/2226/Marc%20Kaufmann) indicated the function given by the formula in [comment 49](https://forum.azimuthproject.org/discussion/comment/17730/#Comment_17730):

> $$g_!(R) = \bigcap \left\{S \in PX: R \subseteq f_!(S) \right\}.$$

Could someone help me out see why the function above is not well-defined when \\(f\\) is not injective?

(I'm sorry to come back at this puzzle, but it seems I'm missing something very elementary.)

> I believe that if the formulas give well-defined functions, these are necessarily the desired adjoints!

In puzzle 18, we have seen that \\(f_!\\) doesn't have a left adjoint if \\(f\\) is not injective (comments [5](https://forum.azimuthproject.org/discussion/comment/16501/#Comment_16501) and [6](https://forum.azimuthproject.org/discussion/comment/16509/#Comment_16509)).

Does this mean that the function given by the formula is not well-defined?

[Marc Kaufmann](https://forum.azimuthproject.org/profile/2226/Marc%20Kaufmann) indicated the function given by the formula in [comment 49](https://forum.azimuthproject.org/discussion/comment/17730/#Comment_17730):

> $$g_!(R) = \bigcap \left\{S \in PX: R \subseteq f_!(S) \right\}.$$

Could someone help me out see why the function above is not well-defined when \\(f\\) is not injective?

(I'm sorry to come back at this puzzle, but it seems I'm missing something very elementary.)