any 1-cell $X$ the identity will be denoted ${\mbox{\rm id\,}}_X$ or sometimes simply as $1_X$, when space saving is needed. For any 2-category ${{\mathcal B}}$, we shall denote by ${{\mathcal B}}^{{\rm{op}}}$ the 2-category that is obtained from ${{\mathcal B}}$ by reversing 1-cells.

\[initial-2cat\] The unit 2-category $\mathcal{I}$ has a single 0-cell, named $\star$. The monoidal category $\mathcal{I}(\star,\star)$ is the unit monoidal category.

A *pseudofunctor* $(F,\alpha):{{\mathcal B}}\to {{\mathcal B}}'$, consists of a function $F:\operatorname{Obj}({{\mathcal B}})\to \operatorname{Obj}({{\mathcal B}}')$, a family of functors $F:{{\mathcal B}}(A,B)\to {{\mathcal B}}'(F(A),F(B)),$ for each $A, B\in \operatorname{Obj}({{\mathcal B}})$, a collection of isomorphisms $\phi_A: I_{F(A)}\to F(I_A),$ and a family of natural isomorphisms $$\xymatrix{
{{\mathcal B}}(B,C)\times {{\mathcal B}}(A,B) \ar[d]_{F\times F} \ar[rr]^{\circ} & {\ar @{} [d] |{\Uparrow \alpha}}& {{\mathcal B}}(A,C) \ar[d]^{F} \\
{{\mathcal B}}'(F(B),F(C))\times {{\mathcal B}}'(F(A),F(B)) \ar[rr]^-{\circ},&& {{\mathcal B}}'(F(A),F(C))
}$$ for 0-cells $A, B, C$, subject to the usual axioms. A pseudofunctor is called *unital* if $F(I_A)= I_{F(A)}$, for any 0-cell $A$, and the isomorphisms $\phi_A$ are the identities. A pseudofunctor is called a *2-functor* if the associativity isomorphisms $\alpha$ are the identities.

If $F$, $G$ are pseudofunctors, a *pseudonatural transformation* consists of a family of 1-cells $\chi^0_A: F(A)\to G(A)$, $A\in \operatorname{Obj}({{\mathcal B}})$ and isomorphisms

$$\xymatrix{
F(A) \ar[r]^-{F(X)} \ar[d]_{\chi^0_A} & F(B) 