Categories and Homomorphism I understand the notation homomorphism set notation, but I'm a little confused about the necessity of the subnotation. We've used $Hom_C(M,N)$ to represent the homomorphisms from $M$ to $N$ in the category $C$
However, when talking about dual spaces, is this necessary? We write $Hom_k(k,V)$ as the dual space. Isn't the category given? Or am I misunderstanding the subscript $k$?
 A: You mean $\text{Hom}_k(V, k)$. This is used to clarify that we're talking about $k$-linear homomorphisms. In some contexts we might want to talk about $R$-linear homomorphisms where $R$ is some other ring in the picture, so it's useful to do this kind of clarification sometimes. 
A: It is quite common throughout mathematics for similar or identical notations to be used with different meanings in different contexts.  In the context of vector spaces over a field (or more generally, modules over a ring), the field (or ring) $k$ is often written as a subscript on "Hom" to indicate that you are considering $k$-linear maps.  In other contexts, you often subscript a Hom-set in a category with the name of the category.  In either context, it is common to drop the subscript entirely when there is little risk of confusion.
(By the way, the dual space is $\operatorname{Hom}(V,k)$, not $\operatorname{Hom}(k,V)$.)
A: Te above guys beat me to it, so I will try to complement the answers. The dual space is $V^*: = Hom_k(V,k)$. 
In particular, $V,k$ are also abelian groups, so $Hom(V,k)$ could mean a homomorphism of abelian groups, the subscript clarifies which kind of homomorphisms are you talking about, or over which field (ring).
