Reusing the same class in container wiring with different containers

[darrenrahnemoon]

Say I have a class with the following dependency

class Foo:
  dependency: Provide['my_dependency']

Can two containers running in the same process reuse the same class to resolve the dependency to something different?
i.e

class ContainerOne:
    my_dependency = Singleton(SomeDependencyClass)
    wiring_coniguration.... same

class ContainerTwo:
    my_dependency = Singleton(SomeOtherDependencyClass)
    wiring_coniguration.... same

Is Provide['my_dependency'] stashing annotations about what to inject in the class itself or in the container that is crawling through the files to wire things together?