I am having a bit of trouble understanding the free functor, T: Set $\rightarrow$ Grp, where T($A$) = Free group generated by $A$. The question I have in mind is how do I map arrows. In Set, the arrows are just functions from $A$ to $B$, with no regards to any structure whatsoever. But in Grp, the arrows are group morphisms. I cant figure out how to map my arrows in Set to a morphism between groups.
Any hints or insight is deeply appreciated
Cheers