I want my app to be able to copy PDF files from Safari, just like adobe reader can copy them from Safari.
Is this a private API that adobe reader uses? I can't find any guide about such action and by reading apple documentation I became more confused and now I have no idea if I need to use a UIActivityViewController or a UIDocumentInteractionController.
You need to look into the Document Interaction API
You should specifically look into the Registering Your Support of File Types
and Opening Files From Other Apps
sections.
Once you've registered PDF as a file type your app supports, your app will appear in the list that shows when you press the More...
button.
"Adobe Acrobat"
on the right.