Adjoints and Monads

For every object \(X \in C\) and every object \(Y \in D\), there exists a bijection between the set of morphisms from \(F(X)\) to \(Y\) in \(D\) and the set of morphisms from \(X\) to \(G(Y)\) in \(C\).

For every object \(X \in C\) and every object \(Y \in D\), there exists a natural isomorphism between the functor \(F\) and the functor \(G\).

For every object \(X \in C\) and every object \(Y \in D\), there exists a natural transformation from \(F\) to \(G\) and a natural transformation from \(G\) to \(F\) such that their composition is the identity natural transformation.

For every object \(X \in C\) and every object \(Y \in D\), there exists a natural transformation from \(F\) to \(G\) such that its composition with itself is the identity natural transformation.