Julio Song wrote:

> "Left" and "right" only make sense with respect to something, and in Galois connections they are (meant to be) with respect to the given function (i.e. the function whose inverse/adjoint we are supposed to find)

Julio, I'd say you're right on. The one thing I'd add is that people often say things like "This function \\(f\\) is a left adjoint" *without* a reference to any other function, and what they mean is "There exists a function \\(g\\) such that \\(f\\) is the left adjoint of \\(g\\)." This is okay, because \\(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". So, totally free of any other context, we can say things like

"The ceiling function from R to Z is a left adjoint." (It *has* a right adjoint, namely the inclusion Z to R.)

or

"The floor function from R to Z is a right adjoint." (The inclusion Z to R is its left adjoint.)

or

"The inclusion Z to R is both a left and right adjoint." (Its right adjoint is ceiling and its left adjoint is floor.)

> "Left" and "right" only make sense with respect to something, and in Galois connections they are (meant to be) with respect to the given function (i.e. the function whose inverse/adjoint we are supposed to find)

Julio, I'd say you're right on. The one thing I'd add is that people often say things like "This function \\(f\\) is a left adjoint" *without* a reference to any other function, and what they mean is "There exists a function \\(g\\) such that \\(f\\) is the left adjoint of \\(g\\)." This is okay, because \\(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". So, totally free of any other context, we can say things like

"The ceiling function from R to Z is a left adjoint." (It *has* a right adjoint, namely the inclusion Z to R.)

or

"The floor function from R to Z is a right adjoint." (The inclusion Z to R is its left adjoint.)

or

"The inclusion Z to R is both a left and right adjoint." (Its right adjoint is ceiling and its left adjoint is floor.)