Notifications on Edit Post Page

Hello. I have a plugin which captures the ‘publish_post’ and ‘edit_post’ hooks to do some post processing on a post after it’s been submitted.

What I need to do is display a notification on the next page to the editor on if all the post processing was successful or not.

I’ve seen many plugins do this, but I’m new to plugin writing (instead of simply plugin using) and can’t seem to figure out how to do it/what exactly those are called so I can check in the codex.

Any help would be appreciated. I’m sure it’s right there, and I keep overlooking it, or searching on the wrong term, etc.