Given two functors $F, G:\mathcal{C}\rightarrow\mathcal{D}$ and a natural transformation $\alpha: F\Rightarrow G$, let $\beta: G\Rightarrow F$ be another natural transformation with the special property that $\beta_c$ is a retraction of $\alpha_c$ for every object $c$ in $\mathcal{C}$, i.e.~for $\beta_c\circ\alpha_c=id_{Fc}$.
I am curious to know if there is a special name for natural transformations like $\beta$.
Suppose the objects of $\mathcal{C}$ forms a set. Let $\mathcal{D}=\mathrm{Set}$ and $G=F^n$, i.e.~$G(A)=(FA, \dots,FA), G(f)=<Ff,\dots,Ff>$. Consider the diagonal natural transformation: for every $A\in ob(\mathcal{C})$ and $x\in A$, $\alpha_A(x)=(x,\dots,x)$. A natural candidate for a $\beta$ that satisfies the retraction property is some projection map: given $(x_1,\dots,x_n)\in FA^n$, let $\beta_A(x_1,\dots,x_n)=x_1$. I am curious to know if these are the only $\beta$'s that satisfy the retraction property.
More generally, I am interested in the existence of such ``retraction'' natural transformations in the most general setting. Any guidance would be much appreciated.