What are "nearly initial" objects really called?

In Definition 4.1 of their paper Shapely monads and analytic functors, Richard Garner and Tom Hirschowitz call such an object a "Galois object".

An earlier reference than Garner–Hirschowitz's paper is Tholen's MacNeille completion of concrete categories with local properties (1979), in which these objects are called quasi-initial (Definition 1.1), defined as those objects that are weakly initial and prequasi-initial (which is exactly the automorphism condition you describe).

On reflection, maybe I do have some terminological suggestions, though I'm not sure how much I like them.

• Model theorists would call a weakly terminal object "universal", so you might call a weakly initial object "co-universal". Bleh.

• Model theorists would call an object with the dual of your automorphism property "homogeneous", so you might use the term "co-homogeneous". Maybe a little less bleh.

• The term "saturated" for model theorists is roughly equivalent to universal + homogeneous, i.e. to "nearly terminal". So you might use the term "co-saturated". This strikes me as a little dangerous because the correspondence in the dual case is not exact.