Let them have their way once in a while: your relationship with them is more important than being right.
Proving them wrong will make them feel beaten and will affect your relationship - it also means you are potentially starting an argument, which will be costly for you too.
How important is it, really, to be right on this particular occasion?
If it’s critical for the project then go ahead and correct them, but if it’s a small issue consider letting them get away with it – if it doesn’t really matter to be right at least you’re winning on the relationship front.
And you can always correct them at a later date: your authority will only be strengthened when you show that this wasn’t a point important enough to be argued at the time.