In exercises 1.3i and 1.3 ii of Riehl's Category Theory they ask "What is a functor between groups, regarded as one-object categories?" and "What is a functor between preorders, regarded as categories?" respectively.
In the case of groups, are we viewing the entire group as a category with only 1 identity morphism from the entire group to itself or are we viewing each element of the group as its own category with only the identity morphisms for each element? Regardless, I'm not sure how to elegantly interpret a functor between groups. I was thinking maybe they're group homomorphisms but in my categoric definition, i didn't appeal to the group operation so this doesn't make sense.
Similarly, in the case of preorders, I understand the categoric definition, and I understand a Functor must map the elements in the preorder to elements of another preorder in the same relative order. But I feel like there should be a more elegant way to describe this and I can't seem to figure this out.
I have a similar issue with interpreting natural transformations between parallel pairs of functors in both group and preorders (exercises 1.4ii and 1.4iii).