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2.2.4 Recovering a $2$-Category from its Duskin Nerve

In ยง1.2.2, we proved that the nerve functor

\[ \operatorname{N}_{\bullet }: \operatorname{Cat}\rightarrow \operatorname{Set_{\Delta }} \]

is fully faithful. This result generalizes to the setting of $2$-categories:

Theorem 2.2.4.1 (Duskin [MR1897816]). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. Then passage to the Duskin nerve induces a bijection

\[ \{ \text{Strictly unitary lax functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \rightarrow \{ \text{Maps of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$} \} . \]

In other words, the Duskin nerve functor $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{ULax}} \rightarrow \operatorname{Set_{\Delta }}$ is fully faithful.

Remark 2.2.4.2. Combining Theorem 2.2.4.1, Theorem 2.2.2.1, and Remark 2.2.0.3, we see that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ determines a fully faithful embedding from the ordinary category of $(2,1)$-categories (where morphisms are strictly unitary functors in the sense of Definition 2.1.4.9) to the ordinary category of $\infty $-categories (where morphisms are functors in the sense of Definition 1.4.0.1).

Remark 2.2.4.3. In [MR1897816], Duskin proves a stronger version of Theorem 2.2.4.1, which also identifies the essential image of the functor $\operatorname{N}^{\operatorname{D}}_{\bullet }: \operatorname{2Cat}_{\operatorname{ULax}} \rightarrow \operatorname{Set_{\Delta }}$.

Corollary 2.2.4.4. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. Then passage to the Duskin nerve induces a bijection

\[ \{ \text{Strictly unitary functors $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$} \} \rightarrow \{ \text{Maps $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ preserving thin $2$-simplices} \} . \]

Proof of Theorem 2.2.4.1. By virtue of Example 2.1.6.14, we may assume without loss of generality that the $2$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are strictly unitary (this assumption will simplify some of the notation in what follows). Let $U: \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ be a map of simplicial sets. Then:

  • Each object $X$ of $\operatorname{\mathcal{C}}$ can be identified with a vertex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ (Example 2.2.1.13), whose image under $U$ is a vertex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$. This vertex can be identified with an object of $\operatorname{\mathcal{D}}$, which we denote by $U_0(X)$.

  • Each $1$-morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ can be identified with an edge of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ (Example 2.2.1.14), whose image under $U$ is an edge of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$. This edge can be identified with a $1$-morphism of $\operatorname{\mathcal{D}}$, which we will denote by $U_1(f): U_0(X) \rightarrow U_0(Y)$.

  • Let $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ be $1$-morphisms of $\operatorname{\mathcal{C}}$, and let $\gamma : g \circ f \Rightarrow h$ be a $2$-morphism of $\operatorname{\mathcal{C}}$. The $2$-morphism $\gamma $ determines a $2$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ (Example 2.2.1.15). The image of this $2$-simplex under $U$ is a $2$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$, which we can identify with a $2$-morphism $U_2(\gamma ): U_1(g) \circ U_1(f) \Rightarrow U_1( g \circ f )$ in $\operatorname{\mathcal{D}}$. Beware that this notation is slightly abusive: the $2$-morphism $U_2(\gamma )$ is a priori dependent not only on $\gamma $, but also on the factorization of the source of $\gamma $ as a composition $g \circ f$.

Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a strictly unitary lax functor. Unwinding the definitions, we see that the induced map of simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{D}}(F): \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ coincides with $U$ if and only if the following conditions are satisfied:

$(0)$

For every object $X \in \operatorname{\mathcal{C}}$, we have $F(X) = U_0(X)$ (as objects of $\operatorname{\mathcal{D}}$).

$(1)$

For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we have $F(f) = U_1(f)$ (as $1$-morphisms from $F(X) = U_0(X)$ to $F(Y) = U_0(Y)$ in $\operatorname{\mathcal{D}}$).

$(2)$

For every triple of $1$-morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$ in $\operatorname{\mathcal{C}}$ and every $2$-morphism $\gamma : g \circ f \Rightarrow h$, the $2$-morphism $U_2(\gamma ): U_1(g) \circ U_1(f) \Rightarrow U_1(h)$ of $\operatorname{\mathcal{D}}$ is given by the (vertical) composition

\[ U_1(g) \circ U_1(f) = F(g) \circ F(f) \xRightarrow { \mu _{g,f} } F( g \circ f) \xRightarrow { F(\gamma )} F( h ) = U_1(h), \]

Let us note two special cases of condition $(2)$. Taking $h = g \circ f$ and $\gamma : g \circ f \Rightarrow h$ to be the identity $2$-morphism, we obtain the following:

$(2_0)$

For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ of $\operatorname{\mathcal{C}}$, the composition constraint $\mu _{g,f}: F(g) \circ F(f) \Rightarrow F(g \circ f)$ coincides with the $2$-morphism $U_2( \operatorname{id}_{g \circ f} )$.

Taking $g$ to be the identity morphism $\operatorname{id}_{Y}: Y \rightarrow Y$ and invoking our assumption that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are strictly unitary, we also obtain:

$(2_1)$

For every pair of $1$-morphisms $f,h: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ and every $2$-morphism $\gamma : f \Rightarrow h$, we have

\[ U_2( \gamma ) = F( \gamma ) \mu _{ \operatorname{id}_{Y}, f} = F(\gamma ) \]

(here the second identity follows from Remark 2.1.4.11, since the $2$-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are strictly unitary).

We wish to show that there is a unique strictly unitary lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ satisfying conditions $(0)$, $(1)$, and $(2)$. The uniqueness is clear: by virtue of the analysis above, the functor $F$ must be given on objects, $1$-morphisms, and $2$-morphisms of $\operatorname{\mathcal{C}}$ by the formulae

\[ F(X) = U_0(X) \quad \quad F(f) = U_1(f) \quad \quad F(\gamma ) = U_2(\gamma ) \]

(where, in the third formula, we identify the domain of each $2$-morphism $\gamma : f \Rightarrow h$ in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ with the composition $\operatorname{id}_{Y} \circ f$), and the composition constraint $\mu _{g,f}: F(g) \circ F(f) \Rightarrow F(g \circ f)$ must be given by $\mu _{g,f} = U_2( \operatorname{id}_{g \circ f})$. To complete the proof, it will suffice to show that these formulae supply a well-defined lax functor $F: \operatorname{\mathcal{C}}\Rightarrow \operatorname{\mathcal{D}}$, and that $F$ satisfies condition $(2)$ above (note that $F$ satisfies conditions $(0)$ and $(1)$ by construction).

We first show that $F$ satisfies condition $(2)$. Suppose we are given a triple of $1$-morphisms $f: X \rightarrow Y$, $g: Y \rightarrow Z$, and $h: X \rightarrow Z$, together with a $2$-morphism $\gamma : g \circ f \Rightarrow h$ in the $2$-category $\operatorname{\mathcal{C}}$. Consider the map $\partial \Delta ^3 \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ represented by the pair of diagrams

\[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{ g } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\operatorname{id}_{g \circ f}} & Z \ar [dr]^{ \operatorname{id}_ Z } \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\gamma } & \\ X \ar [ur]^{ f} \ar [urr]_{g \circ f } \ar [rrr]^{h} & & & Z } \]

\[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{g } \ar [drr]_{ g } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\gamma } & Z \ar [dr]^{\operatorname{id}_ Z } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\operatorname{id}_ g} & \\ X \ar [ur]^{f} \ar [rrr]^{h} & & & Z } \]

(see Example 2.2.1.16). Using the identity $\alpha _{ \operatorname{id}_ Z, g, f} = \operatorname{id}_{ g \circ f}$ (Example 2.1.1.12), we see that these diagrams satisfy the compatibility condition of Example 2.2.1.16), and can therefore be regarded as a $3$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$. Applying the map of simplicial sets $U$, we deduce that the diagrams

\[ \xymatrix@C =100pt@R=50pt{ & F(Y) \ar [r]^{ F(g) } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\mu _{g,f}} & F(Z) \ar [dr]^{ \operatorname{id}_{F(Z)} } \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{F(\gamma )} & \\ F(X) \ar [ur]^{ F(f)} \ar [urr]_{F(g \circ f) } \ar [rrr]^{F(h)} & & & F(Z) } \]

\[ \xymatrix@C =100pt@R=50pt{ & F(Y) \ar [r]^{F(g) } \ar [drr]_{ F(g) } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{U_2(\gamma )} & F(Z) \ar [dr]^{\operatorname{id}_{F(Z)} } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\operatorname{id}_{ F(g) }} & \\ F(X) \ar [ur]^{F(f)} \ar [rrr]^{F(h)} & & & F(Z) } \]

determine a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$: that is, we have a commutative diagram

\[ \xymatrix@C =10pt{ & \operatorname{id}_{F(Z)} \circ (F(g) \circ F(f) ) \ar@ {=>}[dl]_-{\mu _{g,f} } \ar@ {=>}[rr]^{\alpha _{\operatorname{id}_{F(Z)},F(g),F(f)} } & & (\operatorname{id}_{F(Z)} \circ F(g) ) \circ F(f) \ar@ {=>}[dr]^-{ \operatorname{id}} & \\ \operatorname{id}_ Z \circ F(g \circ f) \ar@ {=>}[drr]^{ F(\gamma ) } & & & & F(g) \circ F(f) \ar@ {=>}[dll]_{ U_2(\gamma ) } \\ & & F(h). & & } \]

By virtue of Example 2.1.1.12, we see that this is equivalent to the identity $U_2(\gamma ) = F(\gamma ) \mu _{g,f}$ asserted by $(2)$.

Note that from condition $(2)$, we can deduce that $F$ satisfies the dual of condition $(2_1)$: that is, for every $2$-morphism $\gamma : g \Rightarrow h$ in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, we have $F(\gamma ) = U_2(\gamma )$, where the right hand side is computed by regarding $\gamma $ as a $2$-morphism with domain $g \circ \operatorname{id}_{X}$. It follows that the construction of $F$ from $U$ is invariant under the operation of replacing $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ by the opposite $2$-categories $\operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (this will be useful in what follows, since it reduces the number of identities that we need to check).

We now show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the construction of $F$ on $1$-morphisms and $2$-morphisms determines a functor $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Y) )$. For this, we must establish the following:

  • For each $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we have $F( \operatorname{id}_{f} ) = \operatorname{id}_{ F(f)}$ (as $2$-morphisms from $F(f)$ to itself in $\operatorname{\mathcal{D}}$). By definition, this is equivalent to the identity $U_2( \operatorname{id}_{f} ) = \operatorname{id}_{ F(f)}$, which follows from the compatibility of the map $U: \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ with the degeneracy operators

    \[ s_1: \operatorname{N}_{1}^{\operatorname{D}}(\operatorname{\mathcal{C}}) \Rightarrow \operatorname{N}_{2}^{\operatorname{D}}(\operatorname{\mathcal{C}}) \quad \quad s_1: \operatorname{N}_{1}^{\operatorname{D}}(\operatorname{\mathcal{D}}) \Rightarrow \operatorname{N}_{2}^{\operatorname{D}}(\operatorname{\mathcal{D}}). \]
  • For every triple of $1$-morphisms $f,g,h: X \Rightarrow Y$ in $\operatorname{\mathcal{C}}$ and every pair of $2$-morphisms $\gamma : f \Rightarrow g$, $\delta : g \Rightarrow h$, we have $F(\delta \gamma ) = F(\delta ) F(\gamma )$. To prove this, consider the map $\partial \Delta ^3 \rightarrow \operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$ represented by the pair of diagrams

    \[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{ \operatorname{id}_ Y } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\gamma } & Y \ar [dr]^{ \operatorname{id}_ Y } \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\delta } & \\ X \ar [ur]^{ f} \ar [urr]_{ g } \ar [rrr]^{h} & & & Y } \]
    \[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{ \operatorname{id}_ Y} \ar [drr]_{ \operatorname{id}_ Y } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\delta \gamma } & Y \ar [dr]^{ \operatorname{id}_ Y } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{ \operatorname{id}_{ \operatorname{id}_ Y} } & \\ X \ar [ur]^{f} \ar [rrr]^{h} & & & Y } \]

    (see Example 2.2.1.16). It follows from Example 2.1.1.12 that the associativity constraint $\alpha _{ \operatorname{id}_ Y, \operatorname{id}_{Y}, f }$ is the identity, so that the diagrams above satisfy the compatibility condition of Example 2.2.1.16 and therefore determine a $3$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$. Applying the map of simplicial sets $U$, we deduce that there exists a $3$-simplex of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{\mathcal{D}}}$ whose boundary is given by the diagrams

    \[ \xymatrix@C =70pt@R=50pt{ & F(Y) \ar [r]^{ \operatorname{id}_{F(Y)} } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{F(\gamma ) } & F(Y) \ar [dr]^{ \operatorname{id}_{F(Y)} } \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{F(\delta ) } & \\ F(X) \ar [ur]^{ F(f)} \ar [urr]_{ F(G) } \ar [rrr]^{F(h)} & & & F(Y) } \]
    \[ \xymatrix@C =70pt@R=50pt{ & F(Y) \ar [r]^{ \operatorname{id}_{F(Y)}} \ar [drr]_{ \operatorname{id}_{F(Y)} } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{F(\delta \gamma ) } & F(Y) \ar [dr]^{ \operatorname{id}_{F(Y)} } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{ \operatorname{id}_{ \operatorname{id}_{F(Y)}} } & \\ F(X) \ar [ur]^{F(f)} \ar [rrr]^{F(h)} & & & F(Y). } \]

    Using the criterion of Example 2.2.1.16, we see that this is equivalent to the identity $F(\delta \gamma ) = F(\delta ) F(\gamma )$.

We now show that, for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition constraints $\mu _{g,f}: F(g) \circ F(f) \Rightarrow F(g \circ f)$ depends functorially on $f \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $g \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$. We will argue that for fixed $f$, the construction $g \mapsto \mu _{g,f}$ is functorial; functoriality in $g$ will then follow by symmetry. Suppose we are given a $2$-morphism $\gamma : g \Rightarrow h$ in $\operatorname{\mathcal{C}}$; we wish to show that the diagram $\tau :$

\[ \xymatrix@C =50pt@R=50pt{ F(g) \circ F(f) \ar@ {=>}[r]^{ F(\gamma ) \circ \operatorname{id}_{F(f)} } \ar@ {=>}[d]^{\mu _{g,f} } & F(h) \circ F(f) \ar@ {=>}[d]^{ \mu _{h,f} } \\ F( g \circ f) \ar@ {=>}[r]^{ F( \gamma \circ \operatorname{id}_ f )} & F(h \circ f) } \]

commutes in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) )$. To prove this, we consider the map $\operatorname{\partial }\Delta ^3 \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ represented by the pair of diagrams

\[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{ g } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{\operatorname{id}_{g \circ f}} & Z \ar [dr]^{ \operatorname{id}_ Z} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\gamma \circ \operatorname{id}_ f } & \\ X \ar [ur]^{ f } \ar [urr]_{ g \circ f } \ar [rrr]^{ h \circ f } & & & Z } \]

\[ \xymatrix@C =100pt@R=50pt{ & Y \ar [r]^{ g } \ar [drr]_{ h } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\operatorname{id}_{h \circ f } } & Z \ar [dr]^{ \operatorname{id}_ Z } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\gamma } & \\ X \ar [ur]^{f} \ar [rrr]^{h \circ f} & & & Z. } \]

Using the identity $\alpha _{ \operatorname{id}_{Z}, g, f} = \operatorname{id}_{ g \circ f}$ supplied by Example 2.1.1.12, we see that this diagram defines a $3$-simplex of $\operatorname{N}^{\operatorname{D}}_{\bullet }(\operatorname{\mathcal{C}})$. Applying the map of simplicial sets $U$, we deduce that there is a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ whose boundary is represented by the pair of diagrams

\[ \xymatrix@C =100pt@R=50pt{ & F(Y) \ar [r]^{ F(g) } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{ \mu _{g,f} } & F(Z) \ar [dr]^{ \operatorname{id}_{F(Z)}} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{F(\gamma \circ \operatorname{id}_ f) } & \\ F(X) \ar [ur]^{ F(f) } \ar [urr]_{ F(g \circ f) } \ar [rrr]^{ F(h \circ f) } & & & F(Z) } \]

\[ \xymatrix@C =100pt@R=50pt{ & F(Y) \ar [r]^{ F(g) } \ar [drr]_{ F(h) } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\mu _{h,f} } & F(Z) \ar [dr]^{ \operatorname{id}_{F(Z)} } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{F(\gamma )} & \\ F(X) \ar [ur]^{F(f)} \ar [rrr]^{F(h \circ f)} & & & F(Z). } \]

This translates to the commutativity of the diagram

\[ \xymatrix@C =0pt{ & \operatorname{id}_{F(Z)} \circ (F(g) \circ F(f) ) \ar@ {=>}[dl]_-{ \mu _{g,f} } \ar@ {=>}[rr]^{\alpha _{ \operatorname{id}_{F(Z)}, F(g), F(f) } } & & (\operatorname{id}_{F(Z)} \circ F(g)) \circ F(f) \ar@ {=>}[dr]^-{ F(\gamma ) } & \\ \operatorname{id}_{F(Z)} \circ F(g \circ f) \ar@ {=>}[drr]^{ F(\gamma \circ \operatorname{id}_{f}) } & & & & F(h) \circ F(f) \ar@ {=>}[dll]_{ \mu _{h,f} } \\ & & F(h \circ f), & & } \]

which (again by virtue of Example 2.1.1.12) is equivalent to the commutativity of the diagram $\tau $.

To complete the proof, it will suffice to show that $F$ and $\mu $ satisfy conditions $(a)$, $(b)$, and $(c)$ of Definition 2.1.4.3. Condition $(a)$ is immediate from the construction, and $(b)$ follows by symmetry. To verify $(c)$, suppose we are given a triple of composable $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in the $2$-category $\operatorname{\mathcal{C}}$. Consider the $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}})$ represented by the pair of diagrams

\[ \xymatrix@C =100pt@R=50pt{ & X \ar [r]^{ g } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{ \operatorname{id}_{g \circ f} } & Z \ar [dr]^{ h} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{\alpha _{h,g,f} } & \\ W \ar [ur]^{ f} \ar [urr]_{ g \circ f } \ar [rrr]^{(h \circ g) \circ f} & & & Z } \]

\[ \xymatrix@C =100pt@R=50pt{ & X \ar [r]^{ g } \ar [drr]_{ h \circ g } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\operatorname{id}_{(h \circ g) \circ f}} & Y \ar [dr]^{ h } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\operatorname{id}_{h \circ g}} & \\ W \ar [ur]^{f} \ar [rrr]^{(h \circ g) \circ f} & & & Z. } \]

Applying $U$, we obtain a $3$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{D}})$ represented by the pair of diagrams

\[ \xymatrix@C =90pt@R=50pt{ & F(X) \ar [r]^{ F(g) } \ar@ {=>}[]+<5pt,-10pt>;+<15pt,-25pt>^-{ \mu _{g,f} } & F(Z) \ar [dr]^{ F(h)} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>^{ U_2(\alpha _{h,g,f}) } & \\ F(W) \ar [ur]^{ F(f)} \ar [urr]_{ F(g \circ f) } \ar [rrr]^{F((h \circ g) \circ f)} & & & F(Z) } \]

\[ \xymatrix@C =90pt@R=50pt{ & F(X) \ar [r]^{ F(g) } \ar [drr]_{ F(h \circ g) } \ar@ {=>}[]+<20pt,-20pt>;+<45pt,-45pt>_{\mu _{h \circ g, f}} & F(Y) \ar [dr]^{ F(h) } \ar@ {=>}[]+<-5pt,-10pt>;+<-15pt,-25pt>^{\mu _{h,g}} & \\ F(W) \ar [ur]^{F(f)} \ar [rrr]^{F((h \circ g) \circ f)} & & & F(Z), } \]

which is equivalent to the commutativity of the pentagon appearing in the diagram

\[ \xymatrix@R =50pt@C=75pt{ F(h) \circ (F(g) \circ F(f) ) \ar@ {=>}[d]^{ \operatorname{id}_{F(h)} \circ \mu _{g,f} } \ar@ {=>}[r]^-{ \alpha _{F(h), F(g), F(f)} } & (F(h) \circ F(g) ) \circ F(f) \ar@ {=>}[d]^{ \mu _{h,g} \circ \operatorname{id}_{F(f)} } \\ F(h) \circ F( g \circ f) \ar@ {=>}[d]^{\mu _{h, g\circ f}} \ar@ {=>}[dr]^{ U_2( \alpha _{h,g,f} )} & F(h \circ g) \circ F(f) \ar@ {=>}[d]^{ \mu _{h \circ g, f} } \\ F( h \circ (g \circ f)) \ar@ {=>}[r]^-{ F( \alpha _{h,g,f})} & F( (h \circ g) \circ f) } \]

in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(W), F(Z) )$. Since the triangle on the lower left commutes by virtue of $(2)$, it follows that the outer cycle of the diagram commutes, as desired. $\square$