In one of the first Chapters in the Philosophers Stone, Hedwig delivers a note from Hagrid to Harry, in which Hagrid invites Harry to visit him in his hut.
How did Hedwig know that Hagrid wanted to send a note to Harry? Wouldn't it have made more sense for Hagrid to send his note using his own owl, or a random owl he could get hold of?
Note: There is a similar question here, about how Hedwig knew that the Dursleys would send Harry a present: How did Hedwig know the Dursleys would have a Christmas present for Harry?
But the answers there point out that Harry sent a note to the Dursleys via Hedwig, and they then replied using Hedwig. This is not what happened here.