An adjunction is a pair of functors $\mathcal{C} \to_F \mathcal{D}$ and $\mathcal{D} \to_G \mathcal{C}$, with a pair of Natural transformation $\eta: 1_\mathcal{C} \Rightarrow GF$ and $\epsilon: FG \Rightarrow 1_\mathcal{D}$, called the unit and counit, respectively, satisfying the axioms known as triangle identities.

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In this case $F$ is called a Left-adjoint of $G$ and $G$ is called a Right-adjoint of $F$.

https://ncatlab.org/nlab/show/adjunction

Alternative definition in terms of a Natural isomorphism between Hom-sets.

Examples

Free functors are left ajdoints to forgetful functors

Every adjunction gives rise to a monad (Monad) and every monad gives an adjunction