bicategory ${{\mathcal B}}$ is biequivalent to a locally skeletal one ${{\mathcal B}}'$, that is, each of its hom-category is skeletal. Then in ${{\mathcal B}}'$, every 1-cell equivalence is an isomorphism. By Street’s Yoneda lemma for bicategories [@St2 p.117 ], the Yoneda embedding $${{\mathcal B}}'\to \textbf{Bicat}({{\mathcal B}}',\textbf{Cat}): A\mapsto {{\mathcal B}}'^{{\rm{op}}}(A,-),$$ is locally an equivalence. Therefore, ${{\mathcal B}}'$ is biequivalent to ${{\mathcal B}}''$; the full sub-2category of $\textbf{Bicat}({{\mathcal B}}'^{\operatorname{op}},\textbf{Cat})$ determined by the contravariant representables. Since every equivalence in ${{\mathcal B}}'$ is an isomorphism, every equivalence in ${{\mathcal B}}''$ is an isomorphism and ${{\mathcal B}}$ is biequivalent to ${{\mathcal B}}''$.

The tricategory of 2-categories
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Given a pair of 2-categories ${{\mathcal B}}$ and ${{\mathcal B}}'$, we can define the *functor 2-category*, [**2Cat**]{}$({{\mathcal B}},{{\mathcal B}}')$, whose 0-cells are pseudofunctors ${{\mathcal B}}\to {{\mathcal B}}'$, whose 1-cells are pseudonatural transformations, and whose 2-cells are modifications. Given 2-categories ${{\mathcal B}}, {{\mathcal B}}'$ and ${{\mathcal B}}''$, we define a pseudo-functor $$\otimes: {\textbf{2Cat}}({{\mathcal B}}',{{\mathcal B}}'')\times {\textbf{2Cat}}({{\mathcal B}},{{\mathcal B}}')\to {\textbf{2Cat}}({{\mathcal B}},{{\mathcal B}}''),$$ called the *tensor product*. The tensor product at the level of pseudofunctors is the composition. The tensor product of pseudonatural transformations is $$\label{tensorp-psnat}
(\xymatrix@C=8pt{ {{\mathcal B}}' \ar@/^0.6pc/[rr]^{G}\ar@{}[rr]|{\downarrow \beta}
\ar@/_0.6pc/[rr]_{ G'} &  & {{\mathcal B}}''}\big)\big(\xymatrix@C=8pt{ {{\mathcal B}}\ar@/^0.6pc/[rr]^{F}\ar@{}[rr]|{\downarrow \alpha}
\ar@/_0.6pc/[rr]_{ 