Almost assuredly a Celeste or a variant.  It came to me from my step mother and she grew up eating this fig on her family's propery in Sunflower, Mississippi.  The mother tree is quite old.  After tasting this one, I found it to be a very early sugar fig that has a tight eye, thick pulp and a slightly figgy flavor as well.  As a family tree, this will always have a home with me.