However, I have my doubts about this equation. 'Socratity ' is, in virtue of this way of expressing, a '...ness' of things (like the ...ness in redness, and redness is something of particular things, this thing's redness, that thing's redness), that is, it is something that is, or can, be attributed to many things. The identical signification ( = having the same significatum [ = that what is signified by the sign] ) of homo and humanitas (man and humanity) (which had to be proved) is here already smuggled into the argument by (having as a premise) the equation Socrates = Socratity.
Moreover, all this is just a semantic argument. If the terms ' man ' and ' humanity ' turn out to point to the same thing, namely the individual, then with this it is still not proved that something like a 'universal essence' does not exist. We only don't have a term for it yet. As regards the suppositum of the term ' Socratity ' it is perhaps better to say that Socrates is an instantiation, not of man, but of something that still lies between man and Socrates, namely Socratity. An this is, according to me, what the term actually does. If Socrates is interpreted as an individual substance, differing from Plato, Peter, etc. with respect to content (which interpretation is probably is correct), then Socrates -- as we know him as a contemporary of Plato -- could be an individual (instance) of the species ' Socrates '. The Essence, corresponding to this species, can then be signified by the term ' Socratity ' (provided we do not consider ' Socrates ' exclusively as proper name). A monovular twin could exemplify all this very well.