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Definition (Strict Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $2$-categories. A strict 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 $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, and $F(\gamma )$ for the value of $F$ on a morphism $\gamma $ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$.

This data is required to satisfy the following compatibility conditions:


For every object $X \in \operatorname{\mathcal{C}}$, we have $\operatorname{id}_{F(X)} = F( \operatorname{id}_ X )$.


For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram of categories

\[ \xymatrix@R =50pt@C=50pt{ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \ar [r]^-{\circ } \ar [d]^{F_{Y,Z} \times F_{X,Y}} & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d]^{F_{X,Z}} \\ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(Y),F(Z)) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}(F(X),F(Y)) \ar [r]^-{ \circ } & \underline{\operatorname{Hom}}_{\operatorname{\mathcal{D}}}( F(X), F(Z) ) } \]

is strictly commutative.


For every object $X \in \operatorname{\mathcal{C}}$, the functor $F_{X,X}$ carries the unit constraint $\upsilon _{X}: \operatorname{id}_{X} \circ \operatorname{id}_{X} \xRightarrow {\sim } \operatorname{id}_{X}$ to the unit constraint $\upsilon _{F(X)}: \operatorname{id}_{F(X)} \circ \operatorname{id}_{F(X)} \xRightarrow {\sim } \operatorname{id}_{F(X)}$.


For every composable triple of $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in $\operatorname{\mathcal{C}}$, we have $F( \alpha _{h,g,f} ) = \alpha _{F(h), F(g), F(f) }$. In other words, $F$ carries the associativity constraints of $\operatorname{\mathcal{C}}$ to the associativity constraints of $\operatorname{\mathcal{D}}$.