# Kerodon

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Definition 6.1.0.2. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors between categories. An adjunction between $F$ and $G$ is a pair $(\eta , \epsilon )$, where $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations satisfying the following compatibility conditions:

$(Z1)$

For each object $C \in \operatorname{\mathcal{C}}$, the composite morphism

$F(C) \xrightarrow { F( \eta _ C) } (F \circ G \circ F)(C) \xrightarrow { \epsilon _{F(C)} } F(C)$

is equal to the identity $\operatorname{id}_{ F(C)}$.

$(Z2)$

For each object $D \in \operatorname{\mathcal{D}}$, the composite morphism

$G(D) \xrightarrow { \eta _{ G(D)} } (G \circ F \circ G)(D) \xrightarrow { G(\epsilon _ D)} G(D)$

is equal to the identity $\operatorname{id}_{ G(D) }$.

If these conditions are satisfied, then we will refer to $\eta$ as the unit of the adjunction $(\eta , \epsilon )$ and to $\epsilon$ as the counit of the adjunction $(\eta , \epsilon )$. In this case, we will say that $(\eta , \epsilon )$ exhibits $F$ as a left adjoint to $G$ and also that it exhibits $G$ as a right adjoint to $F$.