In my category theory text I've come across a lot of questions where my own background is lacking in some topics. I have the following question I'm trying to work through:
Describe nontrivial functors between $\textbf{Set}$ and $[G,\textbf{Set}]$
Naturally, my first thought was the forgetful functor $U:[G,\textbf{Set}] \to \textbf{Set}$ taking the $G$-set to its set and the $G$-equivariant map to its natural map of sets. Of course this functor should have an adjoint, and, without invoking the adjoint functor theorem, I should be able to construct it. But I encounter the problem of defining how the functor $F:\textbf{Set} \to [G,\textbf{Set}]$ should take an arbitrary set map $A \to B$ to a natural transformation, i.e. a $G$-equivariant map. Clearly, not all set maps are equivariant, and I cannot seem to construct one arbitrarily. I haven't had any formal classes or previous experiences with equivariant maps, so how might one go about this?
Given $f: A \rightarrow B$ a set map. We can think it as a $G-$ map as follows: Consider actions of $G$ on $A$ and $B$ are trivial.Then $f$ is $G$-equivariant.