wfs.tex

% -*- root: thesis.tex -*-

\chapter{WEAK FACTORIZATION SYSTEMS}\label{Ch:Wfs}

We will begin by briefly reviewing the notions of functorial factorization, weak factorization system, and algebraic weak factorization system. 

\section{Arrow Categories}

Let $\cat{C}$ be a category. Its arrow category $\cat{C}^2$ is the category whose objects are arrows in $\cat{C}$ and whose morphisms are commutative squares. The arrow category comes with two functors $\dom,\cod\colon \cat{C}^2\to\cat{C}$, along with a natural transformation $\kappa\colon\dom\Rightarrow\cod$. The component of $\kappa$ at an object $f$ of $\cat{C}^2$ is simply $f\colon\dom f\to\cod f$. Moreover, $\cat{C}^2$ satisfies a universal property: there is an equivalence of categories
\begin{equation}\label{Eq:ArrowObject}
	\mathrm{Fun}(2,\mathrm{Fun}(\cat{X},\cat{C}))\simeq\mathrm{Fun}(\cat{X},\cat{C}^2)
\end{equation}
given by composition with $\kappa$. Here, 2 is the ordinal, i.e. the category with two objects and a single non-identity arrow. In other words, $\cat{C}^2$ is the cotensor of $\cat{C}$ with the category 2 in the 2-category $\mathcal{C}\mathrm{at}$.

We will make this universal property more explicit in the next lemma, separating out the 1-dimensional and the 2-dimensional parts of~\eqref{Eq:ArrowObject}:

\begin{lemma}\label{Lem:ArrowObject}
	Let $\cat{C}$ be a category.
	\begin{itemize}
		\item[i)] For any category $\cat{X}$, pair of functors $F,G\colon\cat{X}\to\cat{C}$, and natural transformation $\alpha\colon F\Rightarrow G$, there is a unique functor $\hat{\alpha}\colon\cat{X}\to\cat{C}^2$ such that $\dom\hat{\alpha}=F$, $\cod\hat{\alpha}=G$, and
		\begin{equation}
		\begin{tikzcd}[bend angle=30]
			\cat{X} \rar{\hat{\alpha}} 
			& \cat{C}^2 \rar[bend left][domA]{\dom}
				\rar[bend right][codA,swap]{\cod}
			& \cat{C}
			\twocellA{\kappa}
		\end{tikzcd}
		=
		\begin{tikzcd}[bend angle=30]
			\cat{X} \rar[bend left][domA]{F}
				\rar[bend right][codA,swap]{G}
			& \cat{C}.
			\twocellA{\alpha}
		\end{tikzcd}
		\end{equation}
		\item[ii)] For any functors $F,F',G,G'\colon\cat{X}\to\cat{C}$ and a commutative square of natural transformations
		\[
		\begin{tikzcd}[arrows=Rightarrow]
			F \rar{\gamma} \dar[swap]{\alpha}
			& F' \dar{\beta} \\
			G \rar[swap]{\phi}
			& G',
		\end{tikzcd}
		\]
		there is a unique natural transformation $\eta\colon\hat{\alpha}\to\hat{\beta}$ such that $\dom\eta=\gamma$ and $\cod\eta=\phi$, hence
		\begin{equation}\label{Eq:ArrowObject2}
		\begin{tikzcd}[bend angle=50]
			\cat{X}   \rar[bend left][domA]{F}
				\rar[][codA,domB,description,inner ysep=0]{G}
				\rar[bend right][codB,swap]{G'}
			& \cat{C}
			\twocellA[pos=.45]{\alpha}
			\twocellB[pos=.55]{\phi}
		\end{tikzcd}
		=
		\begin{tikzcd}[bend angle=30]
			\cat{X} \rar[bend left][domA]{\hat{\alpha}}
					\rar[bend right][codA,swap]{\hat{\beta}}
			& \cat{C}^2 \rar[bend left][domB]{\dom}
				\rar[bend right][codB,swap]{\cod}
			& \cat{C}
			\twocellA{\eta}
			\twocellB{\kappa}
		\end{tikzcd}
		=
		\begin{tikzcd}[bend angle=50]
			\cat{X}   \rar[bend left][domA]{F}
				\rar[][codA,domB,description,inner ysep=0]{F'}
				\rar[bend right][codB,swap]{G'}
			& \cat{C}.
			\twocellA[pos=.45]{\gamma}
			\twocellB[pos=.55]{\beta}
		\end{tikzcd}
		\end{equation}
	\end{itemize}
\end{lemma}

\begin{definition}\label{Def:ArrowObject}
	Let $\cat{D}$ be any 2-category. For any object $A$ in $\cat{D}$, the \emph{arrow object} of $A$, if it exists, is an object $A^2$ satisfying the universal property~\eqref{Eq:ArrowObject}. If every object has an arrow object, i.e. if $\cat{D}$ has cotensors by 2, we will say $\cat{D}$ \emph{has arrow objects}.
\end{definition}

In practice, we will work with arrow objects in a 2-category using the two parts of \cref{Lem:ArrowObject}.

Finally, we will record here a simple proposition which will be needed later.

\begin{proposition}\label{Prop:ArrowSimplex}
	Any arrow object $A^2$ has an internal category structure
	\[
	\begin{tikzcd}
		A^3 \rar{c} & A^2 \rar[shift left=1.5ex][]{\dom} \rar[shift right=1.5ex][swap]{\cod} & A \lar[][description]{i}
	\end{tikzcd}
	\]
	where $A^3$ is the pullback of the span
	\[
	\begin{tikzcd}
		A^2 \rar{\cod} & A & A^2 \lar[swap]{\dom}
	\end{tikzcd}
	\]
\end{proposition}
\begin{proof}
	Using the universal property, we define $i$ and $c$ by the equations $\dom i=\id$, $\cod i=\id$, $\kappa i=\id_{\id}$, $\dom c=\dom p_1$, $\cod c=\cod p_2$, and $\kappa c=\kappa p_2 \circ \kappa p_1$, where $p_1$ and $p_2$ are the projections of the pullback.
\end{proof}

\section{Functorial Factorizations}\label{Sec:FuncFact}

\begin{definition}\label{Def:CatFF}
	A functorial factorization on a category $\cat{C}$ consists of a functor $E$ and two natural transformations $\eta$ and $\epsilon$ which factor $\kappa$, as in
	\[
	\begin{tikzcd}[bend angle=30]
		C^2 \rar[bend left][domA]{\dom}
			\rar[bend right][codA,swap]{\cod}
		& C
		\twocellA{\kappa}
	\end{tikzcd}
	=
	\begin{tikzcd}[bend angle=50]
		C^2 \rar[bend left][domA]{\dom}
			\rar[][codA,domB,description]{E}
			\rar[bend right][codB,swap]{\cod}
		& C.
		\twocellA[pos=.45]{\eta}
		\twocellB[pos=.55]{\epsilon}
	\end{tikzcd}
	\]
\end{definition}

This determines for any arrow $f$ in $\cat{C}$ a factorization $f=\epsilon_f\circ\eta_f$. The factorization is natural, meaning that for any morphism $(u,v)\colon f\Rightarrow g$ in $\cat{C}^2$ (i.e. commutative square in $\cat{C}$), the two squares in
\[
\begin{tikzcd}
	\cdot \rar{u} \dar[swap]{\eta_f} & \cdot \dar{\eta_g} \\
	\cdot \rar{E(u,v)} \dar[swap]{\epsilon_f} & \cdot \dar{\epsilon_g} \\
	\cdot \rar[swap]{v} & \cdot
\end{tikzcd}
\]
commute.

A functorial factorization also determines two functors $L,R\colon C^2\to C^2$ such that $\dom L=\dom$, $\cod R=\cod$, $\cod L = \dom R = E$, $\kappa L = \eta$, and $\kappa R=\epsilon$, by the universal property of $C^2$. The components of the factorization of $f$ can then also be referred to as $Lf$ and $Rf$, now thought of as objects in $\cat{C}^2$. There are also two canonical natural transformations, $\vec{\eta}\colon \id\Rightarrow R$ and $\vec{\epsilon}\colon L\Rightarrow\id$, determined by the commuting squares
\[
\begin{tikzcd}[arrows=Rightarrow]
	\dom \rar{\eta} \dar[swap]{\kappa} & E \dar{\epsilon} \\
	\cod \rar[swap]{\id} & \cod
\end{tikzcd}
\qquad\text{and}\qquad
\begin{tikzcd}[arrows=Rightarrow]
	\dom \rar{\id} \dar[swap]{\eta} & \dom \dar{\kappa} \\
	E \rar[swap]{\epsilon} & \cod
\end{tikzcd}
\]
respectively. These make $L$ and $R$ into (co)pointed endofunctors of $\cat{C}^2$.

An algebra for the pointed endofunctor $R$ is an object $f$ in $\cat{C}^2$ equipped with a morphism $\vec{t}\colon Rf\Rightarrow f$, such that $\vec{t}\circ\vec{\eta}_f=\id_f$. Similarly, a coalgebra for the copointed endofunctor $L$ is an $f$ equipped with a morphism $\vec{s}\colon f\Rightarrow Lf$, such that $\vec{\epsilon}_f\circ\vec{s}=\id_f$.

\begin{lemma}
	Let $f\colon X\to Y$ be a morphism in $\cat{C}$. An $R$-algebra structure on $f\in\cat{C}^2$ is precisely a choice of lift $t$ in the square
	\begin{equation}\label{Eq:RAlg}
	\begin{tikzcd}
		X \rar[equals] \dar[swap]{Lf} & X \dar{f} \\
		Ef \rar[swap]{Rf} \urar[dashed]{t} & Y.
	\end{tikzcd}
	\end{equation}
	Dually, an $L$-coalgebra structure on $f$ is precisely a choice of lift $s$ in the square
	\begin{equation}\label{Eq:LAlg}
	\begin{tikzcd}
		X \rar{Lf} \dar[swap]{f} & Ef \dar{Rf} \\
		Y \rar[equals] \urar[dashed]{s} & Y.
	\end{tikzcd}
	\end{equation}

	Moreover, a morphsim $(u,v)\colon f\Rightarrow g$ in $\cat{C}^2$ is a morphism of $R$-algebras if it commutes with the lifts $t$ and $t'$, that is, if in the diagram
	\[
	\begin{tikzcd}
		\cdot 	\rar{u} 
				\dar[swap]{Lf} 
			& \cdot \dar{Lg} \\
		\cdot 	\rar{E(u,v)} 
				\dar[xshift=-0.5ex][swap]{Rf} 
			& \cdot 
				\dar[xshift=0.5ex]{Rg} \\
		\cdot 	\uar[xshift=0.5ex][swap]{t}
				\rar[swap]{v}
			& \cdot \uar[xshift=-0.5ex]{t'}
	\end{tikzcd}
	\]
	we have $t'v=E(u,v)t$.
\end{lemma}

\section{Algebraic Weak Factorization Systems}

To simplify the discussion of weak factorization systems, we will start by introducing a notation. For any two morphisms $l$ and $r$ in $\cat{C}$, write $l\boxslash r$ to mean that for every commutative square
\begin{equation}\label{Eq:LiftingSquare}
\begin{tikzcd}
	\cdot \rar{u} \dar[swap]{l} & \cdot \dar{r} \\
	\cdot \rar[swap]{v} \urar[dashed]{w} & \cdot
\end{tikzcd}
\end{equation}
there exists a lift $w$. In this case, we will say that $l$ has the \emph{left lifting property} with respect to $r$, and that $r$ has the \emph{right lifting property} with respect to $l$. Similarly, for two classes of morphisms $\mathcal{L}$ and $\mathcal{R}$, we will say $\mathcal{L}\boxslash\mathcal{R}$ if $l\boxslash r$ for every $l\in\mathcal{L}$ and $r\in\mathcal{R}$. Finally, we will write $\mathcal{L}^{\boxslash}$ for the class of morphisms having the right lifting property with respect to every morphism of $\mathcal{L}$, and ${}^{\boxslash}\!\mathcal{R}$ for the class of morphisms having the left lifting property with respect to every morphism of $\mathcal{R}$.

\begin{definition}
	A \emph{functorial weak factorization system} on a category $\cat{C}$ consists of a functorial factorization on $\cat{C}$ and two classes $\mathcal{L}$ and $\mathcal{R}$ of morphisms in $\cat{C}$, such that
	\begin{itemize}
		\item for every morphism $f$ in $\cat{C}$, $Lf\in\mathcal{L}$ and $Rf\in\mathcal{R}$,
		\item $\mathcal{L}^{\boxslash}=\mathcal{R}$ and ${}^{\boxslash}\!\mathcal{R}=\mathcal{L}$.
	\end{itemize}
\end{definition}

It a simple and standard proof that the lifting property condition can be replaced by two simpler conditions:

\begin{lemma}
	A functorial weak factorization system can equivalently be defined to be a functorial factorization on $\cat{C}$ and two classes $\mathcal{L}$ and $\mathcal{R}$ of morphisms in $\cat{C}$, such that
	\begin{itemize}
		\item for every morphism $f$ in $\cat{C}$, $Lf\in\mathcal{L}$ and $Rf\in\mathcal{R}$,
		\item $\mathcal{L}\boxslash\mathcal{R}$,
		\item $\mathcal{L}$ and $\mathcal{R}$ are both closed under retracts.
	\end{itemize}
\end{lemma}

In fact, the functorial factorization by itself already determines the two classes of morphisms, with $\mathcal{L}$ the class of morphisms admitting an $L$-coalgebra structure, and $\mathcal{R}$ the class of morphisms admitting an $R$-algebra structure. The lifting properties also follow directly from the functorial factorization, as the next lemma shows.

\begin{lemma}
	For any $L$-coalgebra $(l,s)$ and any $R$-algebra $(r,t)$, there is a canonical choice of lift in the square~\eqref{Eq:LiftingSquare}. Any morphism of $R$-algebras $(u_1,v_1)\colon (r,t)\Rightarrow (r',t')$ and any morphism of $L$-coalgebras $(u_2,v_2)\colon (l',s')\Rightarrow (l,s)$ preserves these canonical choices of lifts.
\end{lemma}
\begin{proof}
	The construction is shown in the diagram
	\begin{equation}\label{Eq:CanonicalLift}
	\begin{tikzcd}
		\cdot 	\rar{u} 
				\dar[swap]{Ll} 
			& \cdot \dar[xshift=0.5ex]{Lr} \\
		\cdot 	\rar[dashed]{E(u,v)} 
				\dar[xshift=-0.5ex][swap]{Rl} 
			& \cdot \uar[xshift=-0.5ex,dashed]{t} 
				\dar{Rr} \\
		\cdot 	\uar[xshift=0.5ex,dashed][swap]{s}
				\rar[swap]{v}
			& \cdot
	\end{tikzcd}
	\end{equation}
	Commutativity of~\eqref{Eq:LiftingSquare} follows immediately from~\eqref{Eq:RAlg} and~\eqref{Eq:LAlg}.

	That a morphism of $R$-algebras preserves these canonical lifts can be seen in the diagram
	\[
	\begin{tikzcd}[column sep=4em]
		\cdot 	\rar{u} 
				\dar[swap]{Ll} 
			& \cdot \rar{u'}
				\dar[xshift=0.5ex]{Lr} 
			& \cdot \dar[xshift=0.5ex]{Lr'} \\
		\cdot 	\rar{E(u,v)} 
				\dar[xshift=-0.5ex][swap]{Rl} 
			& \cdot \rar{E(u',v')} 
				\uar[xshift=-0.5ex]{t} 
				\dar{Rr} 
			& \cdot \uar[xshift=-0.5ex]{t'} 
				\dar{Rr'} \\
		\cdot 	\uar[xshift=0.5ex][swap]{s}
				\rar[swap]{v}
			& \cdot \rar[swap]{v'}
			& \cdot
	\end{tikzcd}
	\]
	noting that $u'tE(u,v)s = t'E(u',v')E(u,v)s = t'E(u'u,v'v)s$.
\end{proof}

This, together with the classical fact that the class of objects admitting a (co)algebra structure for a (co)pointed endofunctor is closed under retracts, gives a third equivalent definition of a functorial weak factorization system.

\begin{lemma}
	A functorial weak factorization system can equivalently be defined to be a functorial factorization on $\cat{C}$ such that
	\begin{itemize}
		\item for every morphism $f$ in $\cat{C}$, $Lf$ admits an $L$-coalgebra structure, and $Rf$ admits an $R$-algebra structure.
	\end{itemize}
\end{lemma}

An $R$-algebra structure on $Rf$ consists of a morphism $\vec{\mu}_f\colon R^2f\to Rf$ in $\cat{C}^2$ such that $\vec{\mu}_f\circ\vec{\eta}_{Rf}=\id_{Rf}$, while an $L$-coalgebra structure on $Lf$ consists of a morphism $\vec{\delta}_f\colon Lf\to L^2f$ such that $\vec{\epsilon}_{Lf}\circ\vec{\delta}_f=\id_{Lf}$. We might hope that it is possible to choose these structures for all $f$ in a natural way, such that they form the components of natural transformations $\vec{\mu}\colon R^2\Rightarrow R$ and $\vec{\delta}\colon L\Rightarrow L^2$.

If we want these choices of lifts to be fully coherent, we should also ask that for any $R$-algebra $(f,t)$, the lift constructed as in~\eqref{Eq:CanonicalLift} for the square~\eqref{Eq:RAlg} is equal to $t$, and similarly for $L$-coalgebras and~\eqref{Eq:LAlg}. Lastly, we should ask that the components $\vec{\mu}_f$ and $\vec{\delta}_f$ are (co)algebra morphisms. These conditions, plus one more ensuring that there is an unambiguous notion of a morphism with both $L$-algebra and $R$-coalgebra structures, lead to the definition of an \emph{algebraic weak factorization system}, first given in \cite{gt:nwfs} (there called \emph{natural} weak factorization systems), and further refined in \cite{garner:nwfs} and \cite{garner:soa}.

\begin{definition}\label{Def:Awfs}
	An \emph{algebraic weak factorization system} on a category $\cat{C}$ consists of a functorial factorization $(L,\vec{\epsilon},R,\vec{\eta})$ together with natural transformations $\vec{\mu}\colon R^2f\Rightarrow Rf$ and $\vec{\delta}\colon L\Rightarrow L^2$, such that
	\begin{itemize}
		\item $\mathbb{R}=(R,\vec{\eta},\vec{\mu})$ is a monad and $\mathbb{L}=(L,\vec{\epsilon},\vec{\delta})$ a comonad on $\cat{C}^2$, and
		\item the natural transformation $\Delta=(\delta,\mu)\colon LR\Rightarrow RL$ determined by the equation $\epsilon L\circ\delta=\mu\circ\eta R \, (=\id_E)$ as in \cref{Lem:ArrowObject} is a distributive law, which in this case reduces to the single condition $\delta\circ\mu = \mu L\circ E\Delta\circ\delta R$.
	\end{itemize}
\end{definition}

% Seeing that an algebraic weak factorization system consists of some underlying data (a functorial factorization) equipped with multiplication and comultiplication structures of sorts, satisfying a compatibility condition between them, sugests that an awfs might be seen as a bialgebra in some way. Indeed, in \cite{garner:nwfs} and \cite{garner:soa} this is made precise, but with a catch: the mutiplication and comultiplication are with respect to \emph{different} monoidal structures.

% There is a generalization of braided monoidal categories called 2-fold monoidal categories, which have two different monoidal structures satisfying a certain higher level compatibility between them, and this is the most general setting (that we know of) in which the notion of bialgebra makes sense.

% One of our primary motivations for this work was to extend this picture of awfs as bialgebras to include the morphisms of awfs introduced in \cite{riehl:nwfs-model}. Recall that a left Quillen functor between model categories is a left-adjoint functor which preserves cofibrations and trivial cofibrations, or equivalently that the right adjoint preserves fibrations and trivial fibrations. Restricting this to just one of the two weak factorization systems making up the model categories, we can define a left Quillen functor between categories equipped with weak factorization systems to be simply a functor which preserves the left class of morphisms.

Just as we saw that a functorial factorization already determines the left and right classes of morphisms, there is a condition we can place on a functor $F$ between categories equipped with functorial factorizations which implies that $F$ preserves the left class.

\begin{definition}\label{Def:FFColax}
	Let $\cat{C}$ and $\cat{D}$ be categories equipped with functorial factorizations $(E_1,\eta_1,\epsilon_1)$ and $(E_2,\eta_2,\epsilon_2)$ respectively. A \emph{colax morphism of functorial factorizations} is a pair $(F,\phi)$ where $F\colon C\to D$ is a functor and $\phi$ is a natural transformation
	\[
	\begin{tikzcd}
		C^2 \rar[][domA]{E_1} \dar[swap]{\hat{F}} & C \dar{F} \\
		D^2 \rar[][codA,swap]{E_2} & D
		\twocellA{\phi}
	\end{tikzcd}
	\]
	such that
	\begin{gather*}
		\begin{tikzcd}[bend angle=50,ampersand replacement=\&]
			C^2 \rar[][domA]{E_1} 
				\dar[swap]{\hat{F}} 
			\& C \dar{F} \\
			D^2 \rar[][codA,domB]{E_2}	
				\rar[bend right][codB,swap]{\cod}
			\& D
			\twocellA{\phi}
			\twocellB{\epsilon_2}
		\end{tikzcd}
		=
		\begin{tikzcd}[bend angle=50,ampersand replacement=\&]
			C^2 \rar[bend left][domA]{E_1} 
				\rar[][codA,domB,swap]{\cod} 
				\dar[swap]{\hat{F}} 
			\& C \dar{F} \\
			D^2 \rar[][codB,swap]{\cod} \& D
			\twocellA{\epsilon_1}
			\twocellB{\id}
		\end{tikzcd}\label{Eq:FFColaxA}
		\\ \shortintertext{and}
		\begin{tikzcd}[bend angle=50,ampersand replacement=\&]
			C^2 \rar[bend left][domA]{\dom} 
				\rar[][codA,domB,swap]{E_1} 
				\dar[swap]{\hat{F}} 
			\& C \dar{F} \\
			D^2 \rar[][codB,swap]{E_2} \& D
			\twocellA{\eta_1}
			\twocellB{\phi}
		\end{tikzcd}
		=
		\begin{tikzcd}[bend angle=50,ampersand replacement=\&]
			C^2 \rar[][domA]{\dom} 
				\dar[swap]{\hat{F}} 
			\& C \dar{F} \\
			D^2 \rar[][codA,domB]{\dom}	
				\rar[bend right][codB,swap]{E_2}
			\& D.
			\twocellA{\id}
			\twocellB{\eta_2}
		\end{tikzcd}\label{Eq:FFColaxB}
	\end{gather*}
	In components, given a morphism $f\colon X\to Y$ in $C$, these two equations simply say that the following diagram commutes:
	\[
	\begin{tikzcd}
		FX \rar{L_2(Ff)} \dar[swap]{F(L_1f)}
			& E_2(Ff) \dar{R_2(Ff)} \\
		F(E_1f) \urar[][inner sep=0pt]{\phi_f} \rar[swap]{F(R_1f)}
			& FY.
	\end{tikzcd}
	\]
\end{definition}

Here, $\hat{F}\colon C^2\to D^2$ is the obvious lift of $F$ to the arrow categories, sending an object $(f\colon X\to Y)$ in $C^2$ to the object $(Ff\colon FX\to FY)$ in $D^2$.

\begin{proposition}\label{Prop:FFColax}
	Let $(F,\phi)$ be a colax morphism of functorial factorizations as above. Then $F$ preserves the left class of morphisms, i.e.~if $f\colon X\to Y$ in $C$ has an $L_1$-coalgebra structure, then $Ff$ has an $L_2$-coalgebra structure.
\end{proposition}
\begin{proof}
	Let $f\colon X\to Y$ be a morphism in the left class in $C$, with $L_1$-coalgebra structure given by the lift $s$ in 
	\[
	\begin{tikzcd}
		X \rar{Lf} \dar[swap]{f} & Ef \dar{Rf} \\
		Y \urar[][inner sep=1pt]{s} \rar[equal] & Y.
	\end{tikzcd}
	\]
	Then $Ff$ has an $L_2$-coalgebra structure given by $\phi_f F(s)$, as shown by the commutativity of the diagram
	\[
	\begin{tikzcd}[column sep=4ex,row sep=4ex]
		FX \ar{rr}{L_2(Ff)} \ar{dd}[swap]{Ff} \ar{dr}[inner sep=0pt]{F(L_1f)}
			&& E_2(Ff) \ar{dd}{R_2(Ff)} \\
		& F(E_1f) \ar{ur}[inner sep=0pt]{\phi_f} \ar{dr}[inner sep=0pt,swap,pos=.4]{F(R_1f)} & \\
		FY \ar{ur}[inner sep=1pt]{Fs} \ar[equal]{rr}
			&& FY
	\end{tikzcd}
	\]
\end{proof}

If $F$ has a right adjoint $G$, then the natural transformation $\phi$ determines a natural transformation $\phi'\colon E_1\hat{G}\to\hat{G}E_2$ which ensures that $G$ preserves the right class of morphisms analagously to~\cref{Prop:FFColax}, and making $G$ what is called a \emph{lax morphism of functorial factorizations}. The relationship between $\phi$ and $\phi'$ is what is known as the mates correspondence (see e.g.~\cite{cgr:mates}), and we say that $\phi'$ is the mate of $\phi$, and vice versa.

It turns out that the mates correspondence is best understood in the context of double categories, which we review in the next section. The theory of mates underlies the definition of adjunctions of awfs, and is the reason why double categories are an essential part of our general framework.