I am looking for examples of non-representable functors, to see how the Yoneda lemma works in these cases.
Here is one: let $\mathbf{C}$ be the category of finite-dimensional Euclidean spaces, with morphisms being isometries (including into higher dimensions). Let $F$ be the functor $\mathbf{C}^{op}\to \mathbf{Set}$ that sends each space $C$ to the set of real-valued functions $C\to \mathbb{R}$ (not necessarily linear or even continuous), and sends each map $f:D \to C$ to the set function sending $g:C\to \mathbb{R}$ to $g \circ f$. We can think of $F$ as defining a presheaf of nonlinear or noncontinuous functions; which means (I think) that $F$ is not representable in $\mathbf{C}$. Yoneda's lemma says that a natural transformation from $Hom(-,C)$ to $F$ is uniquely specified by a section of this presheaf on $C$, i.e. by a specific function $g:C\to \mathbb{R}$.
In this case, $F(C)$ is explicitly a set of functions on the set underlying $C$. Are there any easy-to-understand examples where this is not the case?
Best Answer
Most $\mathbf{Set}$-valued functors aren't representable. You can more or less pick one at random and you'll be good. One approach then is to take a bunch of representable functors and combine them in some way. The result often won't be representable. As a particular case of that, a limit of representable functors being representable is equivalent to the corresponding limit existing in the embedded category. So take any category that doesn't have all limits and a limit it doesn't have gives rise to a limit presheaf that isn't representable.
More generally, all (1-categorical) universal properties correspond to some functor being representable. Again, this leads to plenty of non-representable functors. For example, in the category of vector spaces $\mathsf{Hom}(U\times -,W)$ will typically not be representable because $U\times -$ doesn't have a right adjoint, i.e. the category of vector spaces isn't cartesian closed.