Here is a proof in string diagrams inspired by the proof in Johnson and Yau linked to in Emily's answer.
Read this diagram bottom to top; solid line represents A and dashed lines represent I. Coherence for the underlying monoidal category is assumed. The steps are labeled by 'inv' for invertibility of the braiding, 'nat' for naturality of the braiding, and a hexagon symbol for a hexagon identity.
Update - there is an error in the first diagram, the dashed line should start in a bold dot to represent the unitor - sorry.