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.

The category of automata is discussed in Example 3.3. (3) in:

Adamek, Herrlich, Strecker - Abstract and concrete categories, the Joy of Cats, online.

It also appears in 4K, 5.2, 7.15, 13.13, 15.3, 20H in that book.