Adjoint functors

Remarks

Equivalently, $\varphi$ is a natural isomorphism between the two hom functors $\Hom_{\mathcal{D}}(F -, -)$ and $\Hom_{\mathcal{C}}(-, G -)$ from $\mathcal{C}^{\mathrm{op}} \times \mathcal{D}$ to sets.

Example: let $F$ send a set $X$ to the free group $F(X)$, and let $G$ send a group to its underlying set. A homomorphism $g \colon F(X) \to H$ is determined by its values on the generators, and these values can be any elements of $H$. So $\varphi(g) = g|_X$ is a bijection, and it is natural in $X$ and $H$.

Non-example: the forgetful functor $G$ from groups to sets has no right adjoint. Indeed, a left adjoint preserves colimits, hence the initial object. But $G$ sends the trivial group, which is initial, to a one-point set, which is not initial in sets.

Another standard example is $- \times Y \dashv (-)^Y$ in sets, where $Z^Y$ is the set of maps $Y \to Z$.