Is there a standard notation for a self map? i.e. for a map $$f : U \rightarrow U$$ We can describe $f$ in words as a map on $U$ or a self-map on $U$.

But is there an accepted mathematical notation for this that doesn't involve writing $U$ twice? Maybe a kind of hooked arrow like this:

$$f : U \circlearrowleft$$

Except shaped more like the $\supset$ symbol. I wasn't able to find anything with a quick Google search, and couldn't find anything appropriate on detexify.