Editing composite component

I edited a composite component that is used across multiple pages, however the changes do not show up in the components in the pages? Can someone clarify how to edit a composite component that you created and make the changes work across all pages? I tried the approach of deleting the composite component in the By Me tab so I could save the edited version again however that cannot be deleted without deleting the component that was used to build the composite. Any insight is much appreciated.

This can be filed under RTFM … Core Lesson 4 - Components - Composer