Steps to Reproduce

var s2 = ee.ImageCollection("COPERNICUS/S2_SR_HARMONIZED");
var col = s2.limit(2)

print(col.aggregate_array('MGRS_TILE'))

var change_id = function(img) {
  var custom_name = img.getString('MGRS_TILE').cat('_TESTING')
  return img.set('system:index', custom_name)
}

// calling function directly
var testimage = change_id(col.first())
print("image ID by calling function directly", testimage.id())

// mapping over the collection
var testcol = col.map(change_id)
print("image ID by mapping the function", testcol.first().id())

https://code.earthengine.google.com/cb2c553fa71398e68f9fd4870bd192de

Current Behavior

The property system:index is correctly set when calling the function directly, but it's not set when using the function as a mapping function.

Expected Behavior

The property system:index should be set correctly in both cases.