When I watch cooking shows and they use mayo, mustard or ketchup, I wonder if using the components of those condiments would work. For example
tomato paste in place of ketchup
mustard powder in place of mustard
egg+oil (emulsified) in place of mayo
If you’re replacing ketchup, mix that tomato paste with a generous amount of brown sugar, vinegar, and a little Worcestershire sauce and give it a taste before using it. Same kind of deal with the mustard powder; add vinegar and water to make wet mustard first. I don’t know about the mayo substitute, but I’ve done the other two many times.
Legend-Thanks fopr the answer. I realize I wasn’t totally clear in the OP. I’m thinking mostly when they use those condiments to make another condiment.
That is mayo 
Tri-My idea was that you can make your own, instead of from a jar.
Ketchup brings those flavors to the new condiment, so of you just used tomato paste (let’s say in a quickie Russian dressing), you’d need to contribute the sour vinegar and sweet brown sugar another way.
You can. It will probably taste better than the stuff in jars. You need an electric mixer to do it right, and it just takes a little while.