I am not exactly sure about what you mean with “coming from a company that would dine in a resturant” but Kostenübernahme means that the company pays for the bill, either directly when someone at the table has a company card or someone pays and they get back the money from the company. In both cases you need a bill from the resturant (not just the receipt) that you have to give to your company.
Imagine you have a special ticket to go to a fun amusement park, but you don’t have to pay for it yourself. Instead, your mom or dad tells the amusement park people that they’ll take care of the costs for you. That’s a bit like what “Assumption of costs” means for a company dining in a restaurant.
So, when a company says they’ll assume the costs, it’s like saying, “Don’t worry, we’ll pay for the meal.” After the meal, the restaurant gives them the bill, or how much they need to pay. Then, the company takes that bill and pays for it themselves. They might do this because they’re treating their employees to a meal, or maybe they’re having a business meeting and want to pay for everyone’s food. It’s just a way for the company to say, “We’ve got this!” when it comes to paying for the meal.
In English, “assume” can mean “take on”.
When you “assume the role of manager” at work, that means you take on the job of being manager.
When a company “assumes the cost” at a restaurant, that means they are taking on the cost of the meal. Aka the company will pay the restaurant bill, so the meal is free for the people eating.
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