I am a graduate math student, English is my second language but we have some courses taught in English.

I often see sentences like `There exists *a* unique map/morphism f such that...` (for example, when we define some objects by their universal property)

I am quite puzzled by this. We are talking about something unique, there is one and only one map that has desired properties. So based on what I was taught about articles, I would say that *the* should be used. However, it seems customary to write *a* and I must admit that when reading aloud, saying it with *the* sounds funny.

So is this *a* just a common mistake? And if it is correct, what is the justification for it?