gn.general topology - Something like Yoneda's lemma

This is inspired by The Whitehead for maps question.

Consider two maps f, g: Xto Y which happen to induce the same maps (of discrete spaces) [Z, X] to [Z, Y] for every Z. Does this mean f and g are homotopic?

And what would be the lessons from the answer to this question? I feel like there's something interesting about the way we should ask it.