First, let's examine some word meanings. Nemesis: the inescapable agent of someone's or something's downfall. Also, Nemesis is a goddess: a spirit of divine retribution against those who succumb to hubris, defined specifically as arrogance before the gods.
Cluracan, notably in his behavior in the palace of Morpheus, and his intention (to remove his sister from Morpheus' service) shows hubris. When he gazes into the mirror, it first shows him his own reflection, then the image of Dream/Daniel, who smiles, transforms into a cat, and departs; then it somehow sparks the formation of his own nemesis within him. This indicates that the mirror itself is a mirror of Nemesis, one of the perilous magics of the palace of the Dreaming; Nuala's fear and outrage for her brother, and her instant understanding of what occurred, suggests that this particular peril is not unfamiliar to those who live in the palace.
The agency of the mirror, shown by Dream/Daniel's appearance, sees Cluracan's hubris, and responds with the creation of Cluracan's nemesis (though later, in 'The Wake', we see that Cluracan's nemesis is not so single minded in its intentions, and in fact may play a more ambiguous role in its relationship to him).
But here's where it gets interesting. Daniel/Dream is also Nemesis. If Daniel did not exist, Morpheus would not have been willing to die, Fates or no. It is only because he has a successor that Morpheus, duty-bound as he always has been, is willing to let go. Thus, Dream/Daniel is the nemesis of Morpheus himself, and a mirror of Nemesis, in the realm of Dream, would, by dream-logic, show the manifestation of Dream that reflects its own function.
We know, in Neil Gaiman's world, that prophecy works, and that the future echoes into the past. The very idea of Nemesis implies a knowledge of the future; that the downfall is inescapable, as Morpheus' succession by Daniel is. The mirror shows us this, with Cluracan's experience being only a lesser echo of the Nemesis that permeates the entirety of the Dreaming, the end which has been foreseen, in many different ways, since the beginning of the story.