Factual Authoritative English Definitions of Complex Numbers A The role of the intermediate set ( A ) in any function composition represents the intermediate codomain of the recovery map, and the domain of the extraction map on which id⁡_A closes the bijection. The unique function id⁡A:A→A, id⁡A(a)=a∀a∈A satisfies the two-sided neutral property f∘id⁡A∘g = f∘gf and is a bijection from ( A ) to itself (hence an isomorphism). The “glue set" in the composition f∘g:X→B. ( A ) is the intermediate set in the composition of two functions. It is the exact point where the output of the second operand lands and the input of the first operand begins. In restricted composition: When you restrict the codomain of a second operand (g:Y→Z) and feed its output into a first operand f:Z→W, the intermediate set (Z) carries the identity map id_Z. Inserting id_⁡Z anywhere in the chain yields the original composed map unchanged: f∘id⁡Z∘g = f∘g. Flow of composition: Let ( X, A, B ) be arbitrary sets (of any cardinality). second operand → ( A ) → first operand. Universal quantification over elements of ( A ). ∀a∈A. id⁡A(a)=a Endomorphism + involution: id⁡A−1=id⁡A. g:X→A(second operand / recovery map) and f:A→B(first operand / extraction map) such that the composite map is f∘g:X→B im⁡(g) ⊆ A ⊆ dom⁡(f) ( A ) is a set such that there exist functions g:X→A and f:A→B (for arbitrary sets ( X, B )), simultaneously representing the codomain of the second operand, and the domain of the first operand in a composition. The image of ( g ) becomes the input to ( f ).