>So I'd use \\(\mapsto\\) for the big arrows in your diagram.
>2) A bit more importantly, two of your big arrows have complicated labels that do not describe those processes but rather the _result of applying_ those processes. I would probably write \\(\otimes\\) above the bottom horizontal arrow and \\( \circ \\) next to the right vertical arrow.
Here is a version with the corrections:
![monoidal category string diagram](http://aether.co.kr/images/monoidal_category_string_2.svg)