The Stacks project

Definition 4.29.5. Let $\mathcal{A}$ be a category and let $\mathcal{C}$ be a $2$-category.

1. A functor from an ordinary category into a $2$-category will ignore the $2$-morphisms unless mentioned otherwise. In other words, it will be a “usual” functor into the category formed out of 2-category by forgetting all the 2-morphisms.

2. A weak functor, or a pseudo functor $\varphi $ from $\mathcal{A}$ into the 2-category $\mathcal{C}$ is given by the following data

   1. a map $\varphi : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$,

   2. for every pair $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$, and every morphism $f : x \to y$ a $1$-morphism $\varphi (f) : \varphi (x) \to \varphi (y)$,

   3. for every $x\in \mathop{\mathrm{Ob}}\nolimits (A)$ a $2$-morphism $\alpha _ x : \text{id}_{\varphi (x)} \to \varphi (\text{id}_ x)$, and

   4. for every pair of composable morphisms $f : x \to y$, $g : y \to z$ of $\mathcal{A}$ a $2$-morphism $\alpha _{g, f} : \varphi (g \circ f) \to \varphi (g) \circ \varphi (f)$.

   These data are subject to the following conditions:

   1. the $2$-morphisms $\alpha _ x$ and $\alpha _{g, f}$ are all isomorphisms,

   2. for any morphism $f : x \to y$ in $\mathcal{A}$ we have $\alpha _{\text{id}_ y, f} = \alpha _ y \star \text{id}_{\varphi (f)}$:

      \[ \xymatrix{ \varphi (x) \rrtwocell ^{\varphi (f)}_{\varphi (f)}{\ \ \ \ \text{id}_{\varphi (f)}} & & \varphi (y) \rrtwocell ^{\text{id}_{\varphi (y)}}_{\varphi (\text{id}_ y)}{\alpha _ y} & & \varphi (y) } = \xymatrix{ \varphi (x) \rrtwocell ^{\varphi (f)}_{\varphi (\text{id}_ y) \circ \varphi (f)}{\ \ \ \ \alpha _{\text{id}_ y, f}} & & \varphi (y) } \]

   3. for any morphism $f : x \to y$ in $\mathcal{A}$ we have $\alpha _{f, \text{id}_ x} = \text{id}_{\varphi (f)} \star \alpha _ x$,

   4. for any triple of composable morphisms $f : w \to x$, $g : x \to y$, and $h : y \to z$ of $\mathcal{A}$ we have

      \[ (\text{id}_{\varphi (h)} \star \alpha _{g, f}) \circ \alpha _{h, g \circ f} = (\alpha _{h, g} \star \text{id}_{\varphi (f)}) \circ \alpha _{h \circ g, f} \]

      in other words the following diagram with objects $1$-morphisms and arrows $2$-morphisms commutes

      \[ \xymatrix{ \varphi (h \circ g \circ f) \ar[d]_{\alpha _{h, g \circ f}} \ar[rr]_{\alpha _{h \circ g, f}} & & \varphi (h \circ g) \circ \varphi (f) \ar[d]^{\alpha _{h, g} \star \text{id}_{\varphi (f)}} \\ \varphi (h) \circ \varphi (g \circ f) \ar[rr]^{\text{id}_{\varphi (h)} \star \alpha _{g, f}} & & \varphi (h) \circ \varphi (g) \circ \varphi (f) } \]