If your money is devalued through inflation and your groceries get more expensive, but you can still produce and sell a widget for the same price as before, do you (have to) increase the price of that widget?

Yes, inflation matters a lot, but the psychological part is significant as well (and is conveniently overlooked in basic classical undergrad economics).

A similar thing happened when the Euro was introduced in Europe in 2002: The act of changing the price tags, the confusion of consumers when showing different numbers was an opportunity to hike prices.