In your WordPress multisite, with this plugin you will be allowed to sync from category of a post, to be duplicated to other site. You will have to set in source site and nothing more.
The objective of this plugin is to maintain news of subsites and keep SEO with canonical tag in duplicated entries.
The settings options are in Settings / Publish Multisite.
You can create every pair of category / subsite to make the duplication updated.
