# The Unapologetic Mathematician

## The Strong Yoneda Lemma

We gave a weak, “half-enriched” version of the Yoneda Lemma earlier. Now it’s time to pump it up to a fully-enriched version.

Given a $\mathcal{V}$-functor $F:\mathcal{C}\rightarrow\mathcal{V}$ and an object $K\in\mathcal{C}$, then the functor defines a $\mathcal{V}$-natural map $F_{K,C}:\hom_\mathcal{C}(K,C)\rightarrow F(C)^{F(K)}$. We also have the (ordinary) adjunction $\hom_{\mathcal{V}_0}(\hom_\mathcal{C}(K,C),F(C)^{F(K)})\cong\hom_{\mathcal{V}_0}(F(K),F(C)^{\hom_\mathcal{C}(K,C)})$
and under this adjunction we find $F_{K,C}$ corresponding to a $\mathcal{V}$-natural transformation $\phi_C:F(K)\rightarrow F(C)^{\hom_\mathcal{C}(K,C)}$. Now the strong form of the Yoneda Lemma says that this family is actually the counit of the end $\int_{C\in\mathcal{C}}F(C)^{\hom_\mathcal{C}(K,C)}$, so by the definition of the functor category $\mathcal{V}^\mathcal{C}$ we have an isomorphism in $\mathcal{V}$: $\phi:F(K)\cong\hom_{\mathcal{V}^\mathcal{C}}(\hom_\mathcal{C}(K,\underline{\hphantom{X}}),F)$

So, how do we verify that $F(K)$ is the end in question? Consider any other $\mathcal{V}$-natural family $\alpha_C:X\rightarrow F(C)^{\hom_\mathcal{C}(K,C)}$. Now we run the above adjunction backwards to get a $\mathcal{V}$-natural family $\widetilde{\alpha}_C:\hom_\mathcal{C}(K,C)\rightarrow F(C)^X$. But this is now a $\mathcal{V}$-natural transformation from the functor represented by $K$ to the functor $F(\underline{\hphantom{X}})^X$, and so the weak form of the Yoneda Lemma tells us that $\widetilde{\alpha}_C={1_{F(C)}}^\eta\circ F_{K,C}$ for a unique $\eta:X\rightarrow F(K)$. Running this back through the adjunction says that $\alpha_C=\phi_C\circ\eta$, and the universal property is satisfied.

Let’s hit this isomorphism with the underlying set functor to get a bijection $\hom_{\mathcal{V}_0}(\mathbf{1},F(K))\cong(\mathcal{V}^\mathcal{C})_0(\hom_\mathcal{C}(K,\underline{\hphantom{X}}),F)$. This sends an arrow $\eta:\mathbf{1}\rightarrow F(K)$ to $\phi\circ\eta$, which is a $\mathcal{V}$-natural family with components given by $\phi_C\circ\eta$. But this is exactly the bijection asserted by the weak form of the Yoneda Lemma, so the weaker form is implied by the stronger one.

If we consider the special case where our functor is representable, we find that $\hom_\mathcal{C}(L,K)\cong\hom_{\mathcal{V}^\mathcal{C}}(\hom_\mathcal{C}(K,\underline{\hphantom{X}}),\hom_\mathcal{C}(L,\underline{\hphantom{X}}))$

When $\mathcal{V}^\mathcal{C}$ exists, we can convert the functor $\hom_\mathcal{C}:\mathcal{C}^\mathrm{op}\otimes\mathcal{C}\rightarrow\mathcal{V}$ to a functor $Y:\mathcal{C}^\mathrm{op}\rightarrow\mathcal{V}^\mathcal{C}$ by the exponential adjunction in $\mathcal{V}\mathbf{-Cat}$. By the case of representable functors given above, this Yoneda embedding is fully faithful. From this $\mathcal{V}$-functor we can establish that the Yoneda isomorphism $\phi$ is $\mathcal{V}$-natural in $F$ and $K$.