# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 6.2.1.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty$-categories, and let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ and $\epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$ be natural transformations. Axioms $(Z1)$ and $(Z2)$ of Definition 6.2.1.1 can be restated as follows:

$(Z1)$

There exists a $2$-simplex $\sigma$ of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ with boundary as indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & F \circ G \circ F \ar [dr]^{ \epsilon \circ \operatorname{id}_ F } & \\ F \circ \operatorname{id}_{\operatorname{\mathcal{C}}} \ar [ur]^{ \operatorname{id}_ F \circ \eta } \ar [rr]^{ \operatorname{id}_ F } & & \operatorname{id}_{\operatorname{\mathcal{D}}} \circ F. }$
$(Z2)$

There exists a $2$-simplex $\tau$ of the $\infty$-category $\operatorname{Fun}(\operatorname{\mathcal{D}}, \operatorname{\mathcal{C}})$ with boundary as indicated in the diagram

$\xymatrix@R =50pt@C=50pt{ & G \circ F \circ G \ar [dr]^{ \operatorname{id}_{G} \circ \epsilon } & \\ \operatorname{id}_{\operatorname{\mathcal{C}}} \circ G \ar [ur]^{\eta \circ \operatorname{id}_ G} \ar [rr]^{ \operatorname{id}_ G } & & G \circ \operatorname{id}_{\operatorname{\mathcal{D}}} . }$

In this case, we will say that the $2$-simplices $\sigma$ and $\tau$ witness the axioms $(Z1)$ and $(Z2)$, respectively.