can't find photos library

Typically, the photos library is located in the user/pictures/photo library

From Finder, tap the Go menu at the top of the screen. Hold down the option key so that Library appears in the menu. Tap Library. Locate your name, then Pictures, then Photo Library.

You are chasing rainbows

Neither upgrading to OS X Sierra nor downgrading to El Captain removes anything and certainly nothing from any backup

And Apple disabled nothing - new OSs require updated software and if your software vendors are reputable they will supply the needed upgraded but you have to install them - you choose to upgrade after you read the upgrade documentation and if you upgrade then you have to upgrade what every software needs to be upgraded -

You moved your Photo library to an EHD - what is the format of that drive? How is it connected?

Help is available but you have to do your part and provide details and sane, coherent questions

LN

You are so very welcome - always glad to answer well posed, reasonable questions which include needed details and from people who answer direct questions so correct answers to their questions can be made

Have a really great year!

LN

Due to Apple disabling half my programs when I upgraded to Sierra I had to downgrade my system back to El Capitan.

If you opened your Photos Library while running Sierra, Photos on El Capitan will not be able to open it. Photos on El Capitan can no longer open a library that has been upgraded to Sierra. To be able to use this library with Photos you will have to go back to Sierra to extract the Photos.

And Photos does not show you libraries that are stored on a drive used by Time Machine.

The easiest thing to get access to a library that has been upgraded to Sierra, would be to download the library from iCloud, if you have been using iCloud Photo Library. Syncing a new Lee Capitan Library with the iCloud Photo Library would create a new El Capitan version.

Otherwise check your backup drive directly in the Finder, if your Photos Library is there. Then ctrl-click it , use "Show Package Contents" and copy the Masters folder out of the library. It should contain the original image files of your photos.