Not just that, but even more simply, equality is the intersection of all reflexive relations.
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(This is essentially the adjoint characterization of equality, in the approach where all logical connectives are understood as given by adjoint functors)
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I spoke of equivalence relations rather than reflexive relations because, although reflexive relation is the most general that makes the claim true, the equivalence relation axioms form a complete theory if = is suppressed from the language, and this is the theory satisfied by =.
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How can connectives be understood in terms of adjoint functors?
End of conversation
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