Adjoint objects

Definition 21.1.3. Let \(F\colon C \to D\) be a functor and let \(Y\) be an object of \(D\).

(1)

An object \(Z\) of \(C\) is called a right adjoint object to \(Y\) under \(F\) if it comes equipped with a map \(\epsilon _Y\colon FZ \to Y\) such that for every other object \(X\) of \(C\) the induced map \[ \Hom _C(X,Z) \xrightarrow {F} \Hom _D(FX,FZ) \xrightarrow {\epsilon _Y \circ -} \Hom _D(FX,Y) \] is an equivalence of animae.

(2)

Dually, \(Z\) is called a left adjoint object to \(Y\) under \(F\) if it comes equipped with a map \(\eta _Y\colon Y \to FZ\) such that for every other object \(X\) of \(C\) the induced map \[ \Hom _C(Z,X) \xrightarrow {F} \Hom _D(FZ,FX) \xrightarrow {- \circ \eta _Y} \Hom _D(Y,FX) \] is an equivalence of animae.