Adjoint functors

Definition

Suppose $\mathcal{C}$ and $\mathcal{D}$ are categories. Suppose $\mathcal{F}:\mathcal{C}\to \mathcal{D}$ and $\mathcal{G}:\mathcal{D}\to \mathcal{C}$ are functors. We say that $\mathcal{F}$ is a left-adjoint functor to $\mathcal{G}$ (equivalently, $\mathcal{G}$ is a right-adjoint functor to $\mathcal{F}$) if for every $A\in \mathrm{O}\mathrm{b}\mathcal{C},B\in \mathrm{O}\mathrm{b}\mathcal{D}$, we can define a map:

${\eta }_{AB}:\mathcal{D}(\mathcal{F}A,B)\to \mathcal{C}(A,\mathcal{G}B)$

such that ${\eta }_{AB}$ is a natural transformation in each variable, keeping the other fixed. More precisely:

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