Relation To Adjoint Functors
Suppose (A1, φ1) is an initial morphism from X1 to U and (A2, φ2) is an initial morphism from X2 to U. By the initial property, given any morphism h: X1 → X2 there exists a unique morphism g: A1 → A2 such that the following diagram commutes:
If every object Xi of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φi then define a natural transformation from 1C (the identity functor on C) to UV. The functors (V, U) are then a pair of adjoint functors, with V left-adjoint to U and U right-adjoint to V.
Similar statements apply to the dual situation of terminal morphisms from U. If such morphisms exist for every X in C one obtains a functor V: C → D which is right-adjoint to U (so U is left-adjoint to V).
Indeed, all pairs of adjoint functors arise from universal constructions in this manner. Let F and G be a pair of adjoint functors with unit η and co-unit ε (see the article on adjoint functors for the definitions). Then we have a universal morphism for each object in C and D:
- For each object X in C, (F(X), ηX) is an initial morphism from X to G. That is, for all f: X → G(Y) there exists a unique g: F(X) → Y for which the following diagrams commute.
- For each object Y in D, (G(Y), εY) is a terminal morphism from F to Y. That is, for all g: F(X) → Y there exists a unique f: X → G(Y) for which the following diagrams commute.
Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of C (equivalently, every object of D).
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