A counterexample has to be a counterexample to something.
Pose a concept like "telling the truth"
In this case, the purpose of a counterexample is to show that "telling the truth" is ambiguous with respect to admirable honesty, because it does not represent something only and always admirable.
So the counterexample has to be a story of someone who is telling the truth, but is clearly not admirable.
Such a counterexample story might be: Mary maliciously "tells the truth" about some scandal that happened in Sue's distant past, to cause Sue's boyfriend to break up with her so she can steal the boyfriend from Sue.
The proper response to this counterexample would be an attempt to refine the concept "telling the truth" so it does refer only to admirable honesty. This would make progress toward a refined concept of the essence of admirable honesty.
Sometimes the word "counter-" leads students to mistakenly think that a "counter-example" to the phrase "telling the truth" would show someone doing the opposite of telling the truth, i.e. lying. What would this lead to?
Pose a concept, "telling the truth"
Then comes a story about someone lying rather than telling the truth: Tom maliciously spreads lies, false information about awful things that Jake supposedly did in the past, that never actually occurred. This does not show someone following the rule "tell the truth (as in the case of Mary above), but shows someone doing the opposite of following this rule by lying.
Such a story does not serve the purpose of a counterexample, because it does not show any weakness, ambiguity, or inadequacy in the concept "tell the truth." It tends instead just to confirm the belief in "telling the truth" as a rule invariably describing something admirable, by giving a clear example of obviously bad lying. The purpose of Socratic questioning is to question and undermine confidence in such rules, stimulating a person to try to develop more refined virtue-concepts that do represent something only and always admirable.
Opposite examples can serve some purpose, but their purpose is to help define very specifically what is admirable about admirable V by contrasting admirable V with its non-admirable negative opposites. For example, thinking about "loneliness" as the opposite of romantic love can help stimulate you to think of what specifically is admirable about romantic love by contrast. What exactly is not good about loneliness, and what specific goodness in romantic love is a remedy for this?