Identity is characterized as the initial fixed point of an endofunctor, but the underlying mathematics is the standard initial-algebra theorem and Lambek's lemma.
Bourbaki, Éléments de Mathématique
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Alpay Algebra II: Identity as Fixed-Point Emergence in Categorical Data
Identity is characterized as the initial fixed point of an endofunctor, but the underlying mathematics is the standard initial-algebra theorem and Lambek's lemma.