[Fredrick said](https://forum.azimuthproject.org/discussion/comment/18769/#Comment_18769)

> The error is in treating \\( L \\) and \\( R \\) as if they were left and right ajoints of each other, **they are not**.

I feel this is spot-on but see two similar formulations in Fong & Spivak's book:

1. "We say that \\(f\\) is the _left adjoint_ and \\(g\\) is the _right adjoint_ of the Galois connection." (Definition 1.74)

2. "We say that \\(L\\) _is left adjoint to_ \\(R\\) (and that \\(R\\) _is right adjoint to_ \\(L\\)) ..." (Definition 3.56)

Are these merely loose talks or rather special cases where L and R **are indeed** left/right adjoints of each other?

I think [Owen's comment](https://forum.azimuthproject.org/discussion/comment/18761/#Comment_18761) also addresses this issue, though with a different view (if I have understood correctly)

> [...] \\(f\\) can only be the left adjoint of at most one function, namely its right adjoint if it has one, so "being a left [or right] adjoint" is equivalent to the *property* of "having a right [or left] adjoint".

Anyway, now it looks like there are at least **three** potential reference points for the "left/right" direction of adjunction: (i) the given function, (ii) the (perhaps nonexistent) correct answer, (iii) the other adjoint in the same pair. :-?

> The error is in treating \\( L \\) and \\( R \\) as if they were left and right ajoints of each other, **they are not**.

I feel this is spot-on but see two similar formulations in Fong & Spivak's book:

1. "We say that \\(f\\) is the _left adjoint_ and \\(g\\) is the _right adjoint_ of the Galois connection." (Definition 1.74)

2. "We say that \\(L\\) _is left adjoint to_ \\(R\\) (and that \\(R\\) _is right adjoint to_ \\(L\\)) ..." (Definition 3.56)

Are these merely loose talks or rather special cases where L and R **are indeed** left/right adjoints of each other?

I think [Owen's comment](https://forum.azimuthproject.org/discussion/comment/18761/#Comment_18761) also addresses this issue, though with a different view (if I have understood correctly)

> [...] \\(f\\) can only be the left adjoint of at most one function, namely its right adjoint if it has one, so "being a left [or right] adjoint" is equivalent to the *property* of "having a right [or left] adjoint".

Anyway, now it looks like there are at least **three** potential reference points for the "left/right" direction of adjunction: (i) the given function, (ii) the (perhaps nonexistent) correct answer, (iii) the other adjoint in the same pair. :-?