I am reading the book "Category Theory for Computing Science" about fibrations, and am having trouble understanding what a "homomrophism of opfibrations" is in 12.3.4 FI-2.
FI-2 If $\zeta: (P,\kappa) \rightarrow (P',\kappa')$ is a homomorphism of optifibrations, $F\zeta: F(P,\kappa) \rightarrow F(P',\kappa')$ is the natural transformation ..
There isn't much detail about how $\zeta$ is defined. I am confused because an (op)fibration is $P: E \rightarrow C $ is a functor from categories $E$ to $C$ by definition.
Is there a natural way to define a homomorphism between any two functors? (It seems to me from the quote that there is a natural definition of $\zeta$ which I missed, and the authors then extended $\zeta$ to a natural transformation $F\zeta$.)
A homomorphism preserves some structures. What exactly does a homomorphism from a functor/fibration $P$ to another functor preserve? More specifically, how exactly is the mapping $\zeta$ defined?