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Definition 6.2.1.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories. We will say that a pair of natural transformations $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ are compatible up to homotopy if the following conditions are satisfied:

$(Z1)$

The identity isomorphism $\operatorname{id}_{F}: F \rightarrow F$ is a composition of the natural transformations

\[ F = F \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \xrightarrow { \operatorname{id}_{F} \circ \eta } F \circ G \circ F \quad \quad F \circ G \circ F \xrightarrow { \epsilon \circ \operatorname{id}_{F} } \operatorname{id}_{\operatorname{\mathcal{D}}} \circ F = F \]

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$, in the sense of Definition 1.3.4.1.

$(Z2)$

The identity isomorphism $\operatorname{id}_{G}: G \rightarrow G$ is a composition of the natural transformations

\[ G = \operatorname{id}_{\operatorname{\mathcal{D}}} \circ G \xrightarrow { \eta \circ \operatorname{id}_ G} G \circ F \circ G \quad \quad G \circ F \circ G \xrightarrow { \operatorname{id}_{G} \circ \epsilon } G \circ \operatorname{id}_{\operatorname{\mathcal{D}}} = G \]

in the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$.

We say that a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ is the unit of an adjunction if there exists a natural transformation $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is compatible with $\eta $ up to homotopy. We say that a natural $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ is the counit of an adjunction if there exists a natural transformation $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ which is compatible with $\epsilon $ up to homotopy.