Definition 2.2.4.5 (Lax Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. A lax functor $F$ from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ consists of the following data:
For every object $X \in \operatorname{\mathcal{C}}$, an object $F(X) \in \operatorname{\mathcal{D}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a functor of ordinary categories
\[ F_{X,Y}: \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)). \]We will generally abuse notation by writing $F(f)$ for the value of the functor $F_{X,Y}$ on an object $f$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, an $F(\gamma )$ for the value of $F$ on a morphism $\gamma $ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.
For every object $X \in \operatorname{\mathcal{C}}$, a $2$-morphism $\epsilon _{X}: \operatorname{id}_{ F(X)} \Rightarrow F( \operatorname{id}_ X )$ in the $2$-category $\operatorname{\mathcal{D}}$, which we will refer to as the identity constraint.
For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in the $2$-category $\operatorname{\mathcal{C}}$, a $2$-morphism
\[ \mu _{g,f}: F(g) \circ F(f) \Rightarrow F( g \circ f ), \]which we will refer to as the composition constraint. We require that, if the objects $X$, $Y$, and $Z$ are fixed, then the construction $(g,f) \mapsto \mu _{g,f}$ is functorial: that is, we can regard $\mu $ as a natural transformation of functors as indicated in the diagram
\[ \xymatrix@C =40pt@R=40pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\circ } \ar [d]_{F_{Y,Z} \times F_{X,Y}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F_{X,Z}} \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(Y),F(Z)) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)) \ar [r]^-{ \circ } \ar@ {=>}[]+<30pt,15pt>;+<150pt,45pt>_-{\mu } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]
This data is required to be compatible with the unit and associativity constraints of $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ in the following sense:
- $(a)$
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the left unit constraint $\lambda _{F(f)}$ in $\operatorname{\mathcal{D}}$ is given by the vertical composition
\[ \operatorname{id}_{ F(Y) } \circ F(f) \xRightarrow { \epsilon _{Y} \circ \operatorname{id}_{F(f)} } F( \operatorname{id}_{Y} ) \circ F(f) \xRightarrow { \mu _{ \operatorname{id}_ Y, f} } F( \operatorname{id}_ Y \circ f ) \xRightarrow { F( \lambda _ f) } F(f). \]- $(b)$
For every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, the right unit constraint $\rho _{F(f)}$ in $\operatorname{\mathcal{D}}$ is given by the vertical composition
\[ F(f) \circ \operatorname{id}_{F(X)} \xRightarrow { \operatorname{id}_{F(f)} \circ \epsilon _{X} } F(f) \circ F( \operatorname{id}_ X ) \xRightarrow { \mu _{ f, \operatorname{id}_ X} } F( f \circ \operatorname{id}_ X ) \xRightarrow { F( \rho _ f) } F(f). \]- $(c)$
For every triple of composable $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in the $2$-category $\operatorname{\mathcal{C}}$, we have a commutative diagram
\[ \xymatrix@C =60pt@R=40pt{ F(h) \circ (F(g) \circ F(f) ) \ar@ {=>}[d]^{ \operatorname{id}_{F(h)} \circ \mu _{g,f} } \ar@ {=>}[r]^-{ \alpha _{F(h), F(g), F(f)} } & (F(h) \circ F(g) ) \circ F(f) \ar@ {=>}[d]^{ \mu _{h,g} \circ \operatorname{id}_{F(f)} } \\ F(h) \circ F( g \circ f) \ar@ {=>}[d]^{\mu _{h, g\circ f}} & F(h \circ g) \circ F(f) \ar@ {=>}[d]^{ \mu _{h \circ g, f} } \\ F( h \circ (g \circ f)) \ar@ {=>}[r]^-{ F( \alpha _{h,g,f})} & F( (h \circ g) \circ f) } \]in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(W), F(Z) )$.
A functor from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{D}}$ is a lax functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ with the property that the identity and composition constraints
are isomorphisms.