Is there a way to name the pictures that are collected with Collector for GIS?
According to the documentation, there does not seem to be a way to name the images that are taken using the Collector application. Since the Collector application uses your device's camera, and to my knowledge most devices do not let you name images either, I would say that at this time the answer is no.
Collect data—Collector for ArcGIS | ArcGIS
Thanks for confirming.
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