\ar[d]^{\chi^0_B}\\
G(A)\ar[r]_{G(X)}  {\ar @{} [ru] |{\Downarrow \chi_X}} & G(B)
}$$natural in $X\in {{\mathcal B}}(A,B)$, subject to the usual axioms. If $\chi, \theta$ are pseudonatural transformations, a *modification* from to , consists of a family of 2-cells $\omega_A:\chi^0_A\to \theta^0_A$, such that the diagrams $$\xymatrix{
\chi^0_B\circ F(X) \ar[d]_{\omega_B\circ \operatorname{id}_{F(X)}} \ar[r]^{\chi_X} &  G(X)\circ \chi^0_A \ar[d]^{ \operatorname{id}_{G(X)}\circ \omega_A}\\
\theta^0_B\circ F(X) \ar[r]^{\theta_X} & G(X)\circ \theta^0_A
}$$ commute for all $A\in \operatorname{Obj}({{\mathcal B}})$. This modification will be denoted as $\omega:\chi\Rightarrow \theta$. Given pseudofunctors $F,G:{{\mathcal B}}\to {{\mathcal B}}$, we shall denote $\operatorname{Pseu-Nat}(F,G)$ the category where objects are pseudonatural transformations from $F$ to $G$ and arrows are modifications.

A 1-cell $X\in {{\mathcal B}}(A,B)$ is called an *equivalence* if there exists a 1-cell $Y\in {{\mathcal B}}(B,A)$ such that $X\circ Y\cong I_B$ and $Y\circ X\cong I_A$. We will say that an invertible 1-cell $X$ is an *isomorphism* if there is $X^{*}\in {{\mathcal B}}(B,A)$ such that $X\circ X^{*}= I_B$ and $X^{*}\circ X= I_A$. The next result will be useful later to simplify some proofs.

\[equivalence-1cells-iso\] Every 2-category (or bicategory) is biequivalent to a 2-category where every equivalence 1-cell is an isomorphism.

The proof goes along the lines of [@GPS Theorem 1.4]. Since every category is equivalent to a skeletal one. Every 