The code is generated to get/set properties on your class - the container class in this case. It would set your property to whatever text you use. When the code is executed, your property is set at runtime which sets the property on the TextBox. If you're looking for a way to control the code generation so that setting your property actually serializes the code so that the child TextBox is set, you're not going to find an easy one and I'm not sure it's even possible since VS.NET (or any designer, like SharpDeveloper) is what provides the code serialization implementation. As far as exposing all the properties of a child control, you need to decide what you want to expose to the user. The whole idea of a complex container control is to encapsulate other controls as you require. If you end up exposing all the properties of a child control, then what's the point of encapsulating it in the first place?
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