I want to ask the acceptance problem: Whether a particular deterministic finite automaton accepts a given string can be expressed as a language, $A_{DFA}$. Show $A_{DFA}$ is decidable. The proof in the textbook (Siper page 195) uses two steps:
$M$ = “On input $\langle B,w\rangle$, where $B$ is a DFA and $w$ is a string:
Simulate $B$ on input $w$.
If the simulation ends in an accept state, accept. If it ends in a nonaccepting state, reject .”
What exactly is the meaning of "simulating $B$" in the first step? In my understanding, if $M$ wants simulates multiple DFA $B$, the parameters of $M$ (transition function or acceptance state) need to be changed according to the different input DFA $B$ and input string $w$, since different DFAs may have different structure. In other words, can a TM simulate infinite DFAs? And how it can be done? I know this question may be trivial, but it really bugs me. Any clarification will be appreciated.