Editing Adjoint functors (section)

===Definition via universal morphisms===

By definition, a functor 
<math>F: \mathcal{D} \to \mathcal{C}</math> is a '''left adjoint functor''' if for each object <math>X</math> in <math>\mathcal{C}</math> there exists a [[universal morphism]] 
from <math>F</math> to <math>X</math>. Spelled out, this means that for each object <math>X</math> in <math>\mathcal{C}</math> there exists an object 
<math>G(X)</math> in <math>\mathcal{D}</math> and a morphism <math>\epsilon_X: F(G(X)) \to X</math> such that for every object 
<math>Y</math> in <math>\mathcal{D}</math> and every morphism <math>f: F(Y) \to X</math> there exists a unique morphism 
<math>g: Y \to G(X)</math> with <math>\epsilon_X \circ F(g) = f</math>.

The latter equation is expressed by the following [[commutative diagram]]:
[[File:Definition of the counit of an adjunction.svg|center|Here the counit is a universal morphism.|190px]]

In this situation, one can show that <math>G</math> can be turned into a functor <math>G :\mathcal{C} \to \mathcal{D}</math> in a unique way such that 
<math>\varepsilon_X \circ F(G(f)) = f \circ \varepsilon_{X'}</math> 
for all morphisms <math>f: X' \to X</math> in <math>\mathcal{C}</math>; <math>F</math> is then called a '''left adjoint''' to <math>G</math>.

Similarly, we may define right-adjoint functors. A functor <math>G: \mathcal{C} \to \mathcal{D}</math> is a '''right adjoint functor''' if for each object <math>Y</math> in <math>\mathcal{D}</math>, 
there exists a [[universal morphism]] from <math>Y</math> to <math>G</math>. Spelled out, this means that for each object <math>Y</math> in <math>\mathcal{D}</math>, 
there exists an object <math>F(Y)</math> in <math>C</math> and a morphism <math>\eta_Y: Y \to G(F(Y))</math> such that for every object <math>X</math> in <math>\mathcal{C}</math> 
and every morphism <math>g: Y \to G(X)</math> there exists a unique morphism <math>f: F(Y) \to X</math> with <math>G(f) \circ \eta_Y = g</math>.
[[File:Definition of the unit of an adjunction 1.svg|center|The existence of the unit, a universal morphism, can prove the existence of an adjunction.|190px]]

Again, this <math>F</math> can be uniquely turned into a functor <math>F: \mathcal{D} \to \mathcal{C}</math> such that <math>G(F(g)) \circ \eta_Y = \eta_{Y'} \circ g</math> for <math>g: Y \to Y'</math> a morphism in <math>\mathcal{D}</math>; <math>G</math> is then called a '''right adjoint''' to <math>F</math>.

It is true, as the terminology implies, that <math>F</math> is left adjoint to <math>G</math> if and only if <math>G</math> is right adjoint to <math>F</math>.

These definitions via universal morphisms are often useful for establishing that a given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitively meaningful in that finding a universal morphism is like solving an optimization problem.