So a weird thing about ultraviolet light, like blacklights and UV-C disinfecting light, is that it's invisible. Kinda. And also Kinda.
See there's a distinctive look to blacklights, they look like this.
But it turns out this isn't the ultraviolet that you're seeing.
See, the light produced by the bulb isn't perfectly monochromatic UV. Blacklights are in the UV-A range, which is 315 - 400 nm. The human vision range is 380-nm to 700nm.
There's a little overlap there, between the high UVA and the low end of human vision...
And there's multiple types of ultraviolet lights, but the kinds used in blacklights peak at 370nm or 340nm
so while most of the light coming out of the bulb is in the invisible UV range, some of the light is a bit higher up in the spectrum, which makes it slightly into the visible violet range.
So the light you see from UV lights is either the not-UV light it produces as a side effect, or something else fluorescing.