Representable functors Is the fact of being "representable" only defined for functors $\mathcal{C}\rightarrow\mathbf{Sets}$, or is there some similar concept for other kinds of functors?
For example in an exercise sheet for a course I'm following I am asked to prove that if $G$ is a group and $H$ a subgroup of $G$, then the induced representation $Ind^G_H$ satisfies $Ind^G_H(V) = \hom_H(\mathbb{K}(G),V)$. This looks extremely similar to a representable functor to me, only $Ind^G_H:\mathbf{Vect_\mathbb{K}}\rightarrow \mathbf{Vect_\mathbb{K}}$...
 A: The meaning of presentable is very fluid throughout mathematics. For a group $G$, a representation refers a morphism $G\to End(V)$, thus representing each element in the group $G$ by a linear transformation on some vector space. The Cayley representation theorem from group theory states that every group can be represented faithfully as a subgroup of permutations. Other notions of representations exist (e.g., Brown representability in homotopy theory, matrix representation of a linear transformation any many others). 
The common thread for all representations is that a given object is presented through another kind of object. In category theory, a representation of a functor $F:C\to Set$ is an identification (up to equivalence) of the functor with the hom functor out of a given object $X$, thus $X$ is representing the entire functor. In the context of enriched category theory, say over a nice monoidal category $V$, one can speak of a functor $f:C\to V$ being representable. The case you mention can be restated as saying that the induced representation functor, is represented in $Vect_{\mathbb K}$ by the object $\mathbb K(G)$.
