I am having some trouble understanding what the question is asking me. Here is my understanding. We want to know whether or not there exists a DFA so that when I input a tuple $\langle D,w\rangle$, where $D$ refers to the parameters we define some DFA, and $w$ is a string, it will tell me whether or not $D$ accepts $w$.
I think it is not possible. Any such DFA we construct has a pumping length which differs from the pumping length of any random DFA. However I'm not sure why this contradicts the existence of a universal finite automaton. Any help would be appreciated.