I would like to know if FME has a reader and writer that directly read and write data back to the ArcSDE database.
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It does, yes. Granted, you'll need an Esri license. You can find more details here:
See also https://community.safe.com/s/article/notes-on-fme-and-esri-versions-and-compatibility
@fme_can_do_it , FME does include an ESRI SDE reader/writer that you have access to dependent on licensing.
Here is an article going over the various ESRI readers and writers.
https://community.safe.com/s/article/comparison-of-fme-readers-and-writers-for-sde-and