Exercise 2.1.13 asks:
What can be said about adjunctions between discrete categories?
First of all, what is "adjunction between categories"? Adjunctions between functors were defined on p.41. Is "adjunctions between discrete categories" supposed to mean "adjunctions between functors between discrete categories"?
Secondly, even if we interpret "adjunctions between discrete categories" as I wrote above, what kind of answer is expected from me? I don't think there is any "positive" result. For example, let $Ob(A)=\{a,a',a''\}, Ob(B)=\{b,b'\}$. Consider the functor $F:a,a'\mapsto b; a''\mapsto b'$ and $G:b\mapsto a', b''\mapsto a$. There is no bijection $B(F(a''),b')\cong A(a'',G(b))$.
Best Answer
Yes, that is what's meant. When people say "an adjunction", they usually mean "a pair of functors $(F,G)$ with a natural isomorphism such that blabla"
Secondly, your example is an example of something that isn't an adjunction. In fact, there is a positive result : to get that result you have to answer these :
1) What is a functor between discrete categories ?
2) You have an isomorphism $A(a,G(b)) \simeq B(F(a),b)$. What does this imply with $b=F(a)$, in a discrete category ?
(this 2) is a shadow of the more general notion of the unit of an adjunction, and the counit)