F'} &  & {{\mathcal B}}'}\big)=\big(\xymatrix@C=9pt{ {{\mathcal B}}\ar@/^0.7pc/[rr]^{GF}\ar@{}[rr]|{\downarrow \beta\otimes \alpha}
\ar@/_0.7pc/[rr]_{ G'\!F'} &  & {{\mathcal B}}''}\big),$$ where $$\begin{aligned}
    (\beta \otimes \alpha)_A&=\beta_{F'(A)}\circ G(\alpha_A)\\
    (\beta \otimes \alpha)_X&=(\beta_{F'(X)}\circ \operatorname{id}_{G(\alpha^0_A)})(\operatorname{id}_{\beta^0_{F'(B)}}\circ G(\alpha_X)).\end{aligned}$$ Here, the isomorphisms constraints of the pseudofunctors have been omitted as a space-saving measure. If $\beta':G\to G'$ and $\alpha':F\to F'$ are another pseudonatural transformations and $\omega:\beta\to \beta'$ and $\omega':\alpha\to \alpha'$ are modifications, their tensor product is defined as $\omega \otimes \omega': \beta{{\otimes}}\alpha\to \beta'{{\otimes}}\alpha'$, $(\omega \otimes \omega')_A := \omega_{F'(A)}\circ G(\omega'_A)$, for any 0-cell $A$. If $\alpha:F\to F'$ and $\beta: H\to H'$ are pseudonatural transformations between pseudofunctors $F,F'\in {\textbf{2Cat}}({{\mathcal B}}',{{\mathcal B}}'') ,H,H'\in {\textbf{2Cat}}({{\mathcal B}},{{\mathcal B}}')$, then there is a modification

$$\xymatrix{
&F'H \ar[rd]^{{\mbox{\rm id\,}}_{F'}\otimes \beta}&\\
FH {\ar @{} [rr] |{\Downarrow c_{\alpha,\beta}}} \ar[rd]_{ {\mbox{\rm id\,}}_H\otimes \beta} \ar[ru]^{\alpha\otimes {\mbox{\rm id\,}}_H}&& F'H'\\
&FH' \ar[ru]_{\alpha\otimes {\mbox{\rm id\,}}_{H'}}&
}$$ given by $$\label{comparison-const}
(c_{\alpha,\beta})_A:=\alpha_{\beta_A}^{-1}: F'(\beta_A)\circ \alpha_{H(A)}\to \alpha_{H'(A)}\circ F(\beta_A).$$ This modification is called the *comparison constraint.*

The tensor product is associative only at the level of pseudofunctors, but not for pseudonatural transformations. There exists an associativity constraint

$$\xymatrix{
KHG {\ar @{} [rr] |{\Downarrow a_{\alpha,\beta,\gamma}}} \ar@/^2pc/[rr]^{(\alpha \otimes \beta)\otimes \gamma} \ar@/_2pc/[rr]_{\alpha\otimes (\beta\otimes \gamma)}&& K'H'G'
}$$ for pseudonatural transformations $\alpha:K\to K'$, $\beta:H\to H'$ and $\gamma:G\to G'$. The modification $$(a_{\alpha,\beta,\gamma})_A:\alpha_{F'H'(A)}\circ G(\beta_{H'(A)})\circ GF(\gamma_A)\to\alpha_{F'H'(A)}\circ G(\beta_H'(A)\circ F(\gamma_A))$$ is defined by 