Trying to understand an application of Yoneda's lemma

Question

Suppose I have a category where the functors are $$ (\text{reduced finitely generated commutative k-algebras}) \rightarrow (\text{groups}) $$ that are representable as set valued functors, and the morphisms are natural transformations.

Suppose I have a category where objects are Hopf $k$-algebras, which are reduced, commutative, finitely generated $k$-algebras $A$ with co-multiplication, co-inverse, co-unit such that the co-group axiom holds. And the moprhisms are $k$-algebra morphisms compatiable with the co-group axioms.

With $A \rightarrow Hom_{alg}(A, -)$ these two categories are equivalent by Yoneda's lemma.

I have learned the statement of Yoneda's lemma but I am struggling to see how it can be applied to this situation. I would appreciate any explanation. Thank you.

Answer 1

Removing all the fuzzyness here, you want to prove that for a category $\mathcal C$ with finite coproducts there is an equivalence between the category of co-groups objects in $\mathcal C$ (and co-group morphisms) and the category of representable functors $\mathcal C \to \mathsf{Set}$ that factors through $\mathsf{Grp}$ (and natural transformations).

Lemma. There is an equivalence of category between the group objects in $[\mathcal C,\mathsf{Set}]$ and the functors $\mathcal C\to \mathsf{Set}$ that factors through $\mathsf{Grp}$.

Skecth of proof. Group objects are preserved through evaluation so that any group object of $[\mathcal C,\mathsf{Set}]$ gives rise to a factorisation of it through $\mathsf{Grp}$. The construction just described is fully faithful by definition. It is also essentially surjective: you just have to show that given a functor $F:\mathcal C \to \mathsf{Set}$ that factors through $\mathsf{Grp}$ the collection of all mutplications $(Fc\times Fc \to Fc)_{c\in \mathcal C} $ is a natural transformation, and the same for the collection of units and inversions; this comes from the fact that $F(k:c\to d)$ is not a mere function but a group morphism for every arrow $k$.

Lemma. The group object structures of $\hom(c,-)$ in $[\mathcal C,\mathsf{Set}]$ are equivalent to the co-group structures on $c$ in $\mathcal C$.

Sketch of proof. It boils down to the fact that $\hom(c,-)^n$ is isomorphic to $\hom(\underset{n}{\underbrace{c+\dots+c}},-)$ and then applying Yoneda's lemma.