# Simple examples of non-representable functors

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?

You can build such presheaves by being a little bit silly about things. Observe that if your category $$\mathbb{C}$$ is the category of finite dimensional Euclidean spaces, $$\mathbb{C}$$ does not necessarily admit all colimits. In particular, consider the sequence $$\phi_n:E^{n} \to E^{n+1}$$ given by the mappings, for all $$\mathbf{x} \in E^n$$, $$\phi_n(\mathbf{x}) := (\mathbf{x},0) \in E^{n+1}$$ (so $$\phi_n$$ embeds $$E^n$$ as a subspace of $$E^{n+1}$$ by appending a zero at the end of each vector in $$E^n$$). It is easy to see that the colimit $$E := \lim_{\substack{\longrightarrow \\ n \in \mathbb{N}}} (E^{n},\phi_n)$$ does not exist in $$\mathbb{C}$$ because $$E$$ must be infinite dimensional. However, note that the Yoneda embedding $$\mathbf{y}:\mathbb{C} \to [\mathbb{C}^{\operatorname{op}},\mathbf{Set}]$$ preserves colimits, and the functor category is cocomplete. In this category we have that the colimit $$P(-) := \lim_{\substack{\longrightarrow \\ n \in \mathbb{N}}}(\mathbb{C}(-,E^n),\mathbb{C}(-,\phi_n))$$ exists as a presheaf by the cocompleteness of $$[\mathbb{C}^{\operatorname{op}},\mathbf{Set}]$$. This presheaf, however, is not representable. If it were, it would necessarily need to be represented by the object $$E$$ described above, but since $$E$$ is not an object in your category, this functor is not representable. Essentially $$P$$ is telling you about one type of embedding of any finite dimensional space into some countably infinite Euclidean space, and since $$\mathbb{C}$$ does not admit infinite dimensional spaces as objects, there is no such space in $$\mathbb{C}$$ with which to represent this object.
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.