What is the difference betwen equivalence and isomorphism of functors in categories. I am learning category theory using Basic Category Theory by Tom Leinster as my main source. In the chapter on natural transformations he says that isomorphism of categories is unreasonably strict for the notion of the sameness of two categories. Isomorphism would require functors,
$$
F:A\rightarrow B,G:B\rightarrow A
$$
such that
$$
G\circ F=1_A, F\circ G=1_B
$$
Instead he says that for equivalence we loosen the requirement on these functors to be isomorphic,
$$
G\circ F\cong 1_A,F\circ G\cong 1_B
$$
Then this is better. This section threw me for a loop. I don't understand the difference between the equivalence and the isomorphism statements. Any help clarifying what is trying to be said here is greatly appreciated.

Two functors are said to be isomorphic if there is a natural isomorphism between them, i.e. a natural transformation $\eta: G \circ F \rightarrow 1_A$ such that the components $\eta_X: (G \circ F) (X) \rightarrow 1_A(X)$ are isomorphisms $\forall X \in ob(A)$.
