Customizing Primary Menu to include custom taxonomy description

I’ve a hard time implementing the custom primary menu behaviour.

I’ve already:

• Created a custom post type: products that has a custom
  taxonomy: product_category associated with it.

• Activated and implemented Taxonomy Meta plugin to have the possibility of adding a category image for the product_category taxonomies

The thing I want to achieve is:

• After manually inserting each product_category taxonomy under the Product page in primary menu editing page in admin side, instead of just showing the name of the taxonomy in the client side, I want the menu to show the name, description and the image

Now, I know there’s is a description field in the menu editing page, but what I want is to have that description come from the taxonomy description. Just like the image.

LT;DR
How to customise the primary navigation to show taxonomy description (not the menu item description) next to it’s name? How can I get access to that taxonomy description?

Side note
I’ve created a child theme of Twenty Thirteen but haven’t touched any of it’s functions or menu handling thing etc.