Definition

Let $\mathcal{C}$ and $\mathcal{D}$ be double categories. Denote by $v\mathcal{C}$ (resp. $v\mathcal{D}$) and $h\mathcal{C}$ (resp. $h\mathcal{D}$) the horizontal and vertical categories, respectively, underlying $\mathcal{C}$ (resp. $\mathcal{D}$). Moreover, let us denote by $\vert$ the horizontal composition of squares and by $\frac{\phantom{xx}}{\phantom{xx}}$ the vertical one.

A double functor $F: \mathcal{C} \to \mathcal{D}$ consists of the following data:

• a function $F : Ob \mathcal{C} \to Ob \mathcal{D}$, called the object part,
• a functor $vF : v\mathcal{C} \to v\mathcal{D}$, called the vertical part, coinciding with $F$ on objects,
• a functor $hF : h\mathcal{C} \to h\mathcal{D}$, called the horizontal part, coinciding with $F$ on objects,
• for every square
  $$\array{& X & \overset{f}\rightarrow & Z & \\ g & \downarrow & \alpha &\downarrow & g'\\ &Y & \underset{f'}\rightarrow& W & \\ }$$

  in $\mathcal{C}$, a square

  $$\array{& FX & \overset{hF(f)}\rightarrow & FZ & \\ vF(g) & \downarrow & \square F\alpha &\downarrow & vF(g')\\ &FY & \underset{hF(f')}\rightarrow& FW & \\ }$$

  in $\mathcal{D}$,

obeying the following axioms:

• $\square F(\alpha \vert \beta) = \square F(\alpha) \vert \square F(\beta)$,
• $\square F\left(\frac{\alpha}{\beta}\right) = \frac{\square F(\alpha)}{\square F(\beta)}$,
• For each horizontal map $f : X \to Z$, $\square F(1^h_f) = 1^h_{hF(f)}$ ($\square F$ preserves horizontal identity squares),
• For each vertical map $g : X \to Y$, $\square F(1^v_g) = 1^v_{vF(g)}$ ($\square F$ preserves vertical identity squares)