Given $Hom(X,Y)$, what is the dual, $Hom^{\circ}(X,Y)$

Question

Going through what I thought I had learned about category theory, I find myself a bit unsure about something very elementary: Given a category $C$ with $x, y \in C$, we have the set of morphisms $Hom_C(X,Y)$ (assuming a small category), as well as the dual category, $C^{\circ}$, with its $Hom_{C^{\circ}}(Y,X)$. However, I came across the notation $Hom^{\circ}_C(X,Y)$, and I don't understand it. I mean, is $Hom_C(X,Y)$ even a category? I can't spot the morphisms, at least apart from the identities. Has age finally caught up with me?

Answer 1

As it turns out, I didn't remember what the book actually said - I didn't have the book in front of me when I wrote this, and the author used a different notation: $Hom_C(X,Y)$ is called $C(X,Y)$, the dual of $C$ is $C^{op}$, and what I mistakenly remembered as $Hom^{\circ}_C(X,Y)$ was actually $C^{op}(X,Y)$, which makes a lot more sense, being $Hom_{C^{op}}(X,Y)$. So, mystery solved.