Consider some objects $A$ and $X$ in the category ${\bf C}$, then ${\bf C}[A,X]$ is defined as the type of arrows from $A$ to $X$ (see the definitions in category theory). Given an arrow $f$ from $X$ to $Y$, there is a sensible induced arrow between types ${\bf C}[A,X]$ and ${\bf C}[A,Y]$, namely $g\mapsto f\circ g$.

If we are allowed to create new types as we please, then we could now create a category with objects being the types ${\bf C}[A,X], {\bf C}[A,Y],\dots$ and then use the above arrow mapping to define a functor “${\bf C}[A,-]$”. If ${\bf C}$ is locally small (which means ${\bf C}[A,B]$ can by viewed as a set), then, instead of creating such a new category, we just embed that structure in ${\bf Set}$. We denote the functor from ${\bf C}$ to ${\bf Set}$ by $\mathrm{Hom}(A,-)$.

Note that we write $\mathrm{Hom}(A,B)$ for what we'd usually write as $\mathrm{Hom}_{\bf C}(A,-)\,B$ and we similarly write $\mathrm{Hom}_{\bf C}(A,f)$ for $\mathrm{Hom}_{\bf C}(A,-)(f)$.

Maybe I'll do a seperate entry later, although that's a little tiresome

For all objects $A:\mathrm{Ob}_{\bf C}$, we define the functor $\mathrm{Hom}(-,A)$ with the obvious object mapping and the contravariant arrow mapping $g\mapsto g\circ f$. So an arrow $f$ from $X$ to $Y$ gets mapped to an arrow $\mathrm{Hom}_{\bf C}(f,B)$ between ${\bf C}[Y,A]$ and ${\bf C}[X,A]$.

The picture on the side shows how the functor $\mathrm{Hom}_{\bf C}(-,A)$ maps ${\bf C}$ into ${\bf Set}$.

Here is an example where the image is of interest: Consider the object $U(1)$ in the category of locally compact abelian groups:

For any group $G$, the set $\mathrm{Hom}(G,U(1))$ inherits a group structure from $U(1)$, i.e. $S^1$ with multiplication $\cdot$. This is called the dual group and its elements are called “characters” $\chi$.

For example, consider $\mathbb R$ with addition. We'll see that this group is actually self-dual. Fix a real number $p$ and then e.g. $x\mapsto\mathrm{e}^{ipx}$ is an element $\chi_p$ of $\mathrm{Hom}(\mathbb R,U(1))$. It's a homomorphism, because $\mathrm{e}^{ipa}\cdot\mathrm{e}^{ipb}=\mathrm{e}^{ip(a+b)}$. We can use the multiplication of phases in $U(1)$ to define a group structure on these $\chi$'s by $\chi_p\bullet\chi_q\equiv(x\mapsto\mathrm{e}^{ipx})\bullet(x\mapsto\mathrm{e}^{iqx}):=x\mapsto\mathrm{e}^{i(p+q)x}$. We have a $\chi_p$ for all real numbers $p$ and so, in this case, as promised, we find $\mathrm{Hom}(\mathbb R,U(1))\cong\mathbb R$. As we'll see below, that “new” $\mathbb R$ from constructing the dual is the Fourier space!

Now since we see $\mathrm{Hom}(-,U(1))$ maps groups to groups, we can consider it as an auto-functor in the category of locally compact abelian groups. It turns out that the twice applied functor $\mathrm{Hom}(\mathrm{Hom}(-,U(1)),U(1))$ is naturally isomorphic to the identity functor. That's called the Pontryagin duality. The isomorphism is actually very simple to define: If $\chi\in\mathrm{Hom}(G,U(1))$, then we can map $g\in G$ to $g\mapsto\left(\chi\mapsto\chi(x)\right)$.

Since locally compact groups have the Haar measure, one can form the space of integrable functions $G\to\mathbb C$ and carry it along with these functors. This procedure is the Fourier transform and it really works for all groups in this category! E.g. for $U(1)$ itself, i.e. the periodic interval, we find that the dual group is $\mathbb Z$ and the associated transform is the Fourier series.

Extending this to a category of non-commutative groups is a research subject. Tannakian categories … Grothendieck stuff … we see that it's desirable for the Hom-sets to have their own algebraic structure (in the above, the Hom-set was again a group) and this is where many ideas come from.

A similar construction works if we consider commutative Banach algebra and their maps maps to $\mathbb C$. This is the Gelfand transform business. It reduces to the Fourier transform if we consider the space $L^1(\mathbb R)$ and convolution as multiplication.

Wikipedia: Hom functor, Yoneda lemma

Functor

Locally small category

Set