Downgrading a Photos Library to a Previous Version of Photos
As previously noted, Apple Photos Application is not upward compatible, for example, Photos 3.0 cannot open a Photos Library from Photos 4.0. If you update your MacOS to include a new Photos application, that application will not allow an older version of MacOS and Photos to access your library. I know right, these are YOUR photos not Apples, but it is what Apple says it is. One notable solution is using iCloud photos to help downgrade the library https://discussions.apple.com/docs/DOC-250000297 For me, this was a no thanks because I don't want Apple cloud having my photos and I'm not paying for cloud storage.
So, another option is to (a) make a good backup of your photo library [Emphasized], (b) use the current version of MacOS and Photos to Export your photos (c) open photos, (d) highlight all the photos you want, (e) Click File, (f) Click Export, (g) Click Export "n" items, (h) in the drop down window you can pick all the detail you want included, click Export, (i) a new window opens and you will direct where you want the photos exported. Once exported you can then downgrade your OS and then with that photo app import your photos.
So what if you already downgraded your OS and now that version of photos will not open your library? You have two options: (1) Reinstall the new OS then follow the instructions above to export the photos, (2) use another Mac (a friends) with the new OS: (i) open the photos application, (ii) point the application to your backup copy of your photo library, (iii) highlight all the photos you want, (iv) Click File, (v) Click Export, (vi) Click Export "n" items, (vii) in the drop down window you can pick all the detail you want included, click Export, (viii) a new window opens and you will direct where you want the photos exported. Once exported you can then downgrade your OS and then with that photo app import your photos.
I recommend keeping the exported photos so you never have this problem again. Anytime you connect your phone or camera to your Mac, be sure to place a copy of the new photos into your exported library in addition to adding to photos. I'm a long time Mac user so take my advice at your own risk but it worked for me.