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 $A\in \operatorname {Ob} {\mathcal {C}},B\in \operatorname {Ob} {\mathcal {D}}$, 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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