Cool idea and great sketching! Also Congrats on the Core77 feature.
I’m not a professional chef (although I know a few and have dated one), but I see something I think this concept could use. It seems to me that most restaurant kitchen floors are tile or similar with drains and get very messy when open and hosed down/mopped every night. I think it would be interesting to incorporate a little stand to lift the soft part of the bag off the floor a few inches so water/spills could go around it without making the bag dirty. This way the chef could keep the bag near by while cooking or leave it on the floor somewhere out of the way if there is nowhere else to store it. Research would tell if this is actually something people might want/use, but it just seems to make sense to me.
Heres a quick sketch of whats in my head:
Also in response to what yo and rkuchinsky were saying about the colors: I think they do need some work but I would be hesitant to jump into black/white checkered (or anything that closely resembles chef garb). I think its important that the colors don’t look out of place when the chef is wearing his/her street clothes. Of the chefs that I know, most are hesitant to be in public in the chef cloths and I don’t think they would want a bag that resembled them. Especially for times when traveling to friends/families houses, they most likely would not be wearing chef attire. Maybe take inspiration from the tools they use: high end knifes/kitchen appliances etc. Just a thought.
You can tell it was a busy day at work 
All the best with your idea,
-Evan