Let $k$ be a commutative ring. Let $\text{Alg}$ be the category of commutative $k$-algebras and $\text{CHopf}$ the category of commutative Hopf-algebras. Let us also write $[\text{Alg}, \text{Grp}]$ for the category of representable functors $\text{Alg} \to \text{Grp}$ (the morphisms are natural transformations between such functors).
We have an obvious functor $$\text{CHopf}^{op} \to [\text{Alg}, \text{Grp}]: H \mapsto \text{Hom}_{\text{Alg}}(H,-)$$ since the sets $\text{Hom}_{\text{Alg}}(H,A)$ obtain a group structure via convolution for every algebra $A$. Concretely, the multiplication is defined by $$f \star g := \mu(f \otimes g)\Delta$$ where $\mu$ is the multiplication on $A$ and $\Delta$ is the comultiplication on $H$. On the level of morphisms, the above functor is defined using the Yoneda lemma.
I'm trying to show that this functor is fully faithful. For this, I need to show that if $\lambda: \text{Hom}_{\text{Alg}}(H,-) \to \text{Hom}_{\text{Alg}}(H',-)$ is a natural transformation that consists of group morphisms and where $H,H'$ are commutative Hopf-algebras, then the morphism $$\lambda_H(\text{id}_H) \in \text{Hom}_{\text{Alg}}(H',H)$$ is a Hopf-algebra morphism. In particular, I try to check that $$(\lambda_H(\text{id}_H) \otimes \lambda_H(\text{id}_H))\Delta_{H'}= \Delta_H \lambda_H(\text{id}_H).$$
First of all, is this true? If it is true, how can I show it? I have tried to exploit the fact that $\lambda_H$ is a group morphism but could not conclude.
The condition that $\lambda$ preserves the group structure is equivalent to saying that, for any $A \in \text{Alg}$, the diagram $\require{AMScd}$ \begin{CD} \text{Hom}_{\text{Alg}}(H \otimes_{k} H, A) @>{- \circ \Delta_{H}}>> \text{Hom}_{\text{Alg}}(H,A) \\ @V{- \circ (\lambda_{H}(\text{id}_{H}) \otimes \lambda_{H}(\text{id}_{H}))}VV @VV{- \circ \lambda_{H}(\text{id}_{H})}V\\ \text{Hom}_{\text{Alg}}(H' \otimes_{k} H' ,A) @>>{- \circ \Delta_{H'}}> \text{Hom}_{\text{Alg}}(H',A) \end{CD} commutes ("$- \circ f$" means "precomposition by $f$"). Take $A := H \otimes_{k} H$ above, and consider the image of $\text{id}_{H \otimes_{k} H} \in \text{Hom}_{\text{Alg}}(H \otimes_{k} H,H \otimes_{k} H)$ in $\text{Hom}_{\text{Alg}}(H',H \otimes_{k} H)$; via the bottom-left corner, the image is $(\lambda_{H}(\text{id}_{H}) \otimes \lambda_{H}(\text{id}_{H})) \circ \Delta_{H'}$, and via the top-right corner, the image is $\Delta_{H} \circ \lambda_{H}(\text{id}_{H})$.