So "Epimorphism" is synonymous in reference to categories of sets... I'm not a mathematician, more a philosopher, so in attempting to gain insight from a different term with the same meaning, I wonder if the object as superject (as in Alfred North Whitehead's use of 'subject-superject' as the intercoherent actual entity) is an object by virtue of its self-mapping?
I'd refer you to Whitehead, him being the "real man of mathematics" behind Principia Mathematica, Rusell's extrovertedness seeing the project to completion and assuring the reification of his God-as-logic by outlawing self-reference (subjectivity) in mathematics. Whitehead's metaphysical treatise "Process and Reality" describes the usual use of "subject" as one-sided, for often is meant the actual entity, the abbreviated "subject-superject."
The truth of "philosophical constructions" isn't foreign to the truth of mathematics, just as truth itself is wholly independent upon any realm of discourse. My question regards the unity of these usages to reveal the true meaning beyond the appearances which the appearances indicate.
Just because "surjectivity" is mentioned as the title of an encyclopedic entry, and "superjectivity" isn't, many responders at this forum and others overlook the terms obvious identity, and so disregard what I inform following Whitehead's usage, and so also overlook the transference of meaning to the other -jectivity function I refer to, "subjectivity." It just shows how the followers (rather than authorities) are blinded to the truth (of objectivity as superjectivity in this case) by the manifestation of consensus (objectivity.)
Outside mathematical analysis, and mathematics is not limited to quantity (number,) for instance in synthesis, integration (of the integer) and particularly to the quality of semantics (a very unfortunately neglected part of mathematics,) analogy would clarify what the tendencies of analysis "sever and mutilate" (to paraphrase Spencer-Brown.) So for the semantics of the foundations of mathematics, I pursue a consensus of "superjectivity" in mathematics (in the minds of mathematically inclined individuals) unified with objectivity abstracted from the form of the object or objects we are familiar with, and it would be the infinity which pours meaninig and value from the Spirit (as the infinitesimal singularity, as a constant) to animate the life-world by filling the cups that are forms with the full void which is the absolute infinite.