What does meanthat two finite automata is equivalent? I think that we must define category of finite automata, i.e. we must define $\mathrm{Hom}(A,B)$, where $A,B$ be an arbitrary finite automata. Hence two finite automata $A,B$ are equivalent if there exist isomorphism $f\in\mathrm{Hom}(A,B)$. So, how we can define category of finite automata?



Isomorphism in category of finite automata
You can find these notions, e.g, in the book


Ji.Adamek, V.Trnkova, Automata and Algebras in Categories. Kluwer, 1989,


S.Eilenberg, Automata, languages, and machines, v.A. Academic Press, 1974


and others books. In the first book there is also a more weak notion of equivalence -- automata with the same behavior.


**Addendum:** Let an automaton (with an initial state) has an input alphabet $A$ and an output alphabet $B$. Then for every word $x\in A^\*$ we get in processing the word $y\in B^\*$. The map $f:A^\*\to B^\*, f(x)=y,$ is called a behavior of the automaton. We can consider two automata as equivalent if they have the same behavior.


