[dependency-injection] Dependency Injection vs Factory Pattern

I had the same question as soon as I read about DI and ended up at this post. So finally this is what I understood but please correct me if am wrong.

"Long ago there were little kingdoms with their own governing bodies controlling and taking decisions based on their own written rules. Later on formed a big government eliminating all these little governing bodies which has one set of rules(constitution) and are implemented through courts"

The little kingdoms' governing bodies are "Factories"

The big government is the "Dependency Injector".

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