In this dialog a specific theme is trated: piety. This specific dialogue seems to be related with the process because piety is one of the acusations Socrates faces. Just like most of Plato's works referent to Socrates in this Socrates tries to fiend the universal definision of an ethical term.
In fact, Euthyphro's character is magestially periled. He thinks he is obliged to prosecute his own father beacause he killed one of his workers. What really amazes Socrates is Euthyphro's certainty that he is performing a pious action.