Notation: function codomain is a set of functions

Question

I know that the following notation works: $f:A \rightarrow \mathcal{F}(B,C)$, where $\mathcal{F}(B,C)=\{g|g:B \rightarrow C\}$.

But does the following notation make sense? I mean, is it correct to write: $f: A \rightarrow (B \rightarrow C)$?

Answer 1

Short answer: no. Almost, if you replace the parentheses with curly braces. But don't. Stick with the original notation.

Thanks, Ethan Bolker!