Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 18 / 2 / a / Solution

The covariant form of the Yoneda lemma says that for a locally small category $\mathcal{C}$, a functor $F:\mathcal{C}\to \mathbf{Set}$ and an object $A$, there is a bijection, natural in $A$ and $F$,Explicitly its two directions areFor $u:B\to B^{\prime }$, functoriality gives $F(u){\alpha }_B^x(f)=F(uf)(x)={\alpha }_{B^{\prime }}^x(uf)$, so ${\alpha }^x$ is a natural transformation. Conversely, naturality of $\alpha$ at $f:A\to B$ gives ${\alpha }_B(f)=F(f)({\alpha }_A(1_A))$. This proves that the displayed maps are inverses.

Naturality in $F$ follows because postcomposition by $\theta :F\Rightarrow F^{\prime }$ sends $x$ to ${\theta }_A(x)$. For $u:A^{\prime }\to A$, precomposition of transformations by $\mathcal{C}(u,-)$ sends ${\alpha }^x$ to ${\alpha }^{F(u)(x)}$. This also verifies naturality in the representing object.