Some functorial maps $G\times G\rightarrow G$

Question

Let $G$ be a group. Le diagonal map $\delta:G\rightarrow G\times G$ obviously gives a functorial morphism from the identity functor of $\mathbf{Grp}$ to the functor $P$ sending $G\mapsto G\times G$ (from $\mathbf{Grp}$ to $\mathbf{Grp}$). Is there a map $G\times G\rightarrow G$ (which is not a projection map) giving a functorial morphism from $P$ to the identity functor ? I somehow think that the answer should be no, but can't prove it. Has the existence of such functorial morphisms been studied in category theory in general (or related areas), and if yes then why is this important?

Answer 1

Well, there is the trivial natural transformation which sends everything to the identity. Notice that the multiplication $G \times G \to G$, $(a,b) \mapsto ab$ is not a morphism in $\mathsf{Grp}$ unless $G$ is abelian, so that won't work.

Edit. Let us try to classify all natural transformations $\eta_G : G \times G \to G$. Clearly, $\eta_G(1,-) : G \to G$ is a natural transformation from $\mathrm{id}_{\mathsf{Grp}}$ to itsself. It is a well-known exercise that $\mathrm{End}(\mathrm{id}_{\mathsf{Grp}})=\{1,\mathrm{id}\}$, where $1$ is the trivial transformation (this is essentially because the equation $(xy)^z=x^z y^z$ holds in the free group on $\{x,y\}$ if and only if $z \in \{0,1\}$). Hence, $\eta(1,-)=1$ or $=\mathrm{id}$. The same holds for $\eta(-,1)$. Since $\eta(a,b)=\eta(a,1) \eta(1,b)$, it follows that $\eta=\mathrm{pr}_1$ or $\eta=\mathrm{pr}_2$ or $\eta=1$.