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Warning 6.2.1.20. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories. The following data are essentially equivalent to one another:

  • The datum of a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which admits a right adjoint.

  • The datum of a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ which admits a left adjoint.

  • The datum of a triple $(F,G,\eta )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors and $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction between $F$ and $G$.

  • The datum of a triple $(F,G, \epsilon )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is the counit of an adjunction between $F$ and $G$.

  • The datum of a quintuple $(F, G, \eta , \epsilon , \sigma )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations which are compatible up to homotopy, and $\sigma : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is a $2$-simplex witnessing axiom $(Z1)$ of Definition 6.2.1.1 (see Remark 6.2.1.5).

  • The datum of a quintuple $(F, G, \eta , \epsilon , \tau )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations which are compatible up to homotopy, and $\tau : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ is a $2$-simplex witnessing axiom $(Z2)$ of Definition 6.2.1.1.

The following data are not equivalent to the above (or to each other):

  • The datum of a pair $(F,G)$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors which are adjoint to one another.

  • The datum of a quadruple $(F,G,\eta ,\epsilon )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations which are compatible up to homotopy,

  • The datum of a sextuple $(F, G, \eta , \epsilon , \sigma , \tau )$, where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ are functors, $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are natural transformations, and $\sigma : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ and $\tau : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ are $2$-simplices witnessing axioms $(Z1)$ and $(Z2)$ of Definition 6.2.1.1.

To say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is left adjoint to a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is somewhat imprecise: one should really specify a witness to the adjointness of $F$ and $G$, which can take the form of either a unit $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ or a counit $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$. Given both a unit $\eta $ and a counit $\epsilon $, one can further demand evidence of their compatibility, which can take the form of a $2$-simplex $\sigma : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ witnessing axiom $(Z1)$ or a $2$-simplex $\tau : \Delta ^2 \rightarrow \operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ witnessing axiom $(Z2)$. If one specifies both of the witnesses $\sigma $ and $\tau $, then one can further demand a witness to the compatibility of $\sigma $ with $\tau $; we will return to this point in ยง.