# Kerodon

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Proposition 6.1.2.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f: C \rightarrow D$ and $g: D \rightarrow C$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\eta : \operatorname{id}_{C} \Rightarrow g \circ f$ and $\epsilon : f \circ g \Rightarrow \operatorname{id}_{D}$ be $2$-morphisms. Suppose we are given another object $T \in \operatorname{\mathcal{C}}$ equipped with $1$-morphisms $c: T \rightarrow C$ and $d: T \rightarrow D$, together with $2$-morphisms $\beta : c \Rightarrow g \circ d$ and $\gamma : f \circ c \Rightarrow d$. Then:

$(1)$

If the pair $(\eta , \epsilon )$ satisfies condition $(Z1)$ of Definition 6.1.1.1 and $\beta$ is the right adjunct of $\gamma$, then $\gamma$ is the left adjunct of $\beta$.

$(2)$

If the pair $(\eta , \epsilon )$ satisfies condition $(Z2)$ of Definition 6.1.1.1 and $\gamma$ is the left adjunct of $\beta$, then $\beta$ is the right adjunct of $\gamma$.

Proof. We will prove $(1)$; the proof of $(2)$ follows by applying the same argument in the conjugate $2$-category $\operatorname{\mathcal{C}}^{\operatorname{c}}$. Consider the diagram

$\xymatrix@C =15pt@R=50pt{ f \circ c \ar@ {=>}[r]^-{ \lambda _{c}^{-1} } \ar@ {=>}[d]^{\rho _{f}^{-1}} & f \circ (\operatorname{id}_{C} \circ c) \ar@ {=>}[dl]^{\sim } \ar@ {=>}[r]^-{\eta } & f \circ ((g \circ f) \circ c) \ar@ {=>}[r]^-{\sim } \ar@ {=>}[dl]_{\sim } & f \circ (g \circ (f \circ c))) \ar@ {=>}[r]^-{\gamma } \ar@ {=>}[d]_{\sim } & f \circ (g \circ d) \ar@ {=>}[d]_{\sim } \\ ( f \circ \operatorname{id}_ C) \circ c \ar@ {=>}[r]^-{\eta } & (f \circ (g \circ f) \circ c \ar@ {=>}[r]^-{\sim } & ((f \circ g) \circ f) \circ c \ar@ {=>}[r]^-{\sim } \ar@ {=>}[d]^{\epsilon } & (f \circ g) \circ (f \circ c) \ar@ {=>}[d]^{\epsilon } \ar@ {=>}[r]^-{\gamma } & (f \circ g) \circ d \ar@ {=>}[d]^{\epsilon } \\ & & (\operatorname{id}_ D \circ f) \circ c \ar@ {=>}[r]^-{\sim } \ar@ {=>}[dr]^-{\lambda _{f}} & \operatorname{id}_ D \circ (f \circ c) \ar@ {=>}[r]^-{\gamma } \ar@ {=>}[d]^{\lambda _{f \circ c}} & \operatorname{id}_ D \circ d \ar@ {=>}[d]^{\lambda _ d} \\ & & & f \circ c \ar@ {=>}[r]^-{\gamma } & d }$

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(T,D)$, where the unlabeled morphisms are given by the associativity constraints of $\operatorname{\mathcal{C}}$ (and their inverses). Our assumption that $\beta$ is the right adjunct of $\gamma$ guarantees that the composition along the top line coincides with $\operatorname{id}_{f} \circ \beta$. Consequently, the left adjunct of $\beta$ is the $2$-morphism of $\operatorname{\mathcal{C}}$ given by clockwise composition around the outside of the diagram. On the other hand, axiom $(Z1)$ of Definition 6.1.1.1 guarantees counterclockwise composition around the outside of the diagram coincides with $\gamma$. To complete the proof, it will suffice to show that the diagram commutes. The commutativity of the triangular regions follows from Propositions 2.2.1.14 and 2.2.1.16. The commutativity of the bottom right square follows from the naturality of left unit constraints (Remark 2.2.1.13) and the commutativity of the middle right square from the functoriality of composition. The remaining squares commute by the naturality of the associativity constraints of $\operatorname{\mathcal{C}}$, and the five-sided region commutes by virtue of the pentagon identity. $\square$