in rerum vero numero non facit pluralitatem unitatum repetitio,
vel si de eodem dicam “gladius unus mucro unus ensis unus”.
Therefore in the case of that number by which we number,
the repetition of ones makes a plurality;
but in the number consisting in things
the repetition of ones does not make a plurality,
as, for example, if I say of one and the same thing,
“one sword, one brand, one blade”.
Boethius (Anicius Manlius Severinus Boethius, c. 480–524 A.D.),
De Trinitate (The Trinity Is One God Not Three Gods),
The Theological Tractates, H.F. Stewart, E.K. Rand, S.J. Tester (trans.),
New Edition, Loeb Classical Library, Harvard/Heinemann, 1973.
The use of the concepts of identity and teridentity is not to identify a thing-in-itself with itself, much less twice or thrice over — there is no need and thus no utility in that. I can imagine Peirce asking, on Kantian principles if not entirely on Kantian premisses, “Where is the manifold to be unified?” The manifold requiring unification does not reside in the object but in the phenomena — in the appearances which might have been appearances of different objects but are bound by the indicated identities to be just so many aspects, facets, parts, roles, or signs of one and the same object.
Notice how the various identity concepts actually functioned in the last example, where they had the opportunity to show their behavior in something like their natural habitat.
The use of the teridentity concept in the “giver of a horse to an owner of it” is to say the thing appearing with respect to its quality under an absolute term, “a horse”, the thing appearing with respect to its existence as the correlate of a dyadic relative, “a potential possession”, and the thing appearing with respect to its synthesis as the correlate of a triadic relative, “a gift”, are one and the same thing.