Pseudofunctor

Definition

Let $\mathcal{B}$ and $\mathcal{B}'$ be bicategories with associators $\alpha, \alpha'$ and unitors $\lambda, \rho, \lambda', \rho'$. A pseudofunctor $F \colon \mathcal{B} \to \mathcal{B}'$ consists of:

1. for each object $a$ of $\mathcal{B}$, an object $F(a)$ of $\mathcal{B}'$;
2. for each pair $a, b$, a functor $F_{a,b} \colon \mathcal{B}(a, b) \to \mathcal{B}'(F(a), F(b))$;
3. for composable $f \colon a \to b$ and $g \colon b \to c$, invertible 2-morphisms $\phi_{g,f} \colon F(g) \circ F(f) \Rightarrow F(g \circ f)$, natural in $f$ and $g$;
4. for each object $a$, an invertible 2-morphism $\phi_a \colon 1_{F(a)} \Rightarrow F(1_a)$.

These data satisfy, for all $f \colon a \to b$, $g \colon b \to c$, $h \colon c \to d$:

1. $F(\alpha_{h,g,f}) \cdot \phi_{h \circ g, f} \cdot (\phi_{h,g} \circ 1_{F(f)}) = \phi_{h, g \circ f} \cdot (1_{F(h)} \circ \phi_{g,f}) \cdot \alpha'_{F(h),F(g),F(f)}$;
2. $F(\lambda_f) \cdot \phi_{1_b, f} \cdot (\phi_b \circ 1_{F(f)}) = \lambda'_{F(f)}$;
3. $F(\rho_f) \cdot \phi_{f, 1_a} \cdot (1_{F(f)} \circ \phi_a) = \rho'_{F(f)}$.

Here $\cdot$ is vertical composition of 2-morphisms.