In the style of CWM let $\mathcal A$ and $\mathcal X$ denote categories and let there be an adjunction $\langle F,G,\phi\rangle$ from $\mathcal X$ to $\mathcal A$.
So for every pair $(x,a)$ there is a bijection:$$\phi_{x,a}:\mathcal A(Fx,a)\to\mathcal X(x,Ga)$$
In that situation I got accustomed to $f^{\sharp}$ as a notation for $\phi_{x,a}(f)$ which is also called the right adjunct of arrow $f:Fx\to a$.
For the notation of the inverse I got accustomed to $f^{\flat}$ as left adjunct for arrow $f:x\to Ga$.
I cannot recall anymore where I encountered these notations for the first time, and unfortunately I found on page 36 of this Introduction to Topos theory of Kostecky the same notation but then switched: $f^{\sharp}$ is used there for the left adjunct.
On page 147 of CWM where the Kleisly construction is handled it seems to be confirmed that $f^{\flat}$ should be used for the left adjunct (hence $f^{\sharp}$ for the right).
So MacLane seems to tell me that I am right, and Kostecki that I am wrong…
Off course notation on its own is not a big deal, but I do appreciate uniformity and will not persist in using my notation if it is somehow wrong.
Can someone tell me more about this notation and/or perhaps justify one of the two ways how to use it?
Which of the two ways is commonly used, and is there an underlying motivation for that?
Best Answer
This notation (either way) just isn't that common, and as far as I'm aware from quite a bit of reading of CT material, there is no common notation. My impression is that neither Mac Lane nor Kostecki were trying to set a standard. Some notation was required, particularly if you want to avoid naming the isomorphism, and they picked one. Mac Lane doesn't use this notation in "Sheaves in Geometry and Logic".
Another notation I've seen is $\lceil f\rceil$ and $\lfloor g\rfloor$, though it's not immediately obvious which way should be which. You could make a case for $\lfloor-\rfloor$ being the $\mathsf{Hom}(FA,B)\to\mathsf{Hom}(A,UB)$ direction by reference to the adjunction (Galois connection) for the embedding of integers into reals (which has left adjoint $\lceil-\rceil$ and right adjoint $\lfloor-\rfloor$). Using this same analogy vaguely argues for Kostecki's choice. I would be surprised if one couldn't make up a reasonable seeming analogy for the other direction, though.
Ultimately, whatever notation you use short of just making the isomorphism explicit and applying it, you will need to define it. At that point, you can pick whatever notation you prefer.