monoidal category. See [@MS].

Let ${{\mathcal B}}, {{\mathcal B}}'$ be two 2-categories and $({{\mathcal H}}, \alpha):  {{\mathcal B}}\to{{\mathcal B}}' $ be a unital pseudofunctor. Denote ${{\mathcal Z}}({{\mathcal H}})={\textbf{2Cat}}({{\mathcal B}},{{\mathcal B}}')({{\mathcal H}},{{\mathcal H}})$; the category of pseudonatural transformations of the pseudofunctor ${{\mathcal H}}$. This is a monoidal category with tensor product described in the previous section. Explicitly, objects in ${{\mathcal Z}}({{\mathcal H}})$ are pairs $(V,\sigma)$, where $$V=\{V_A\in {{\mathcal B}}'({{\mathcal H}}(A),{{\mathcal H}}(A)) \text{1-cells, for any} A\in {{\mathcal B}}\},$$ $$\sigma=\{\sigma_X: V_B\circ {{\mathcal H}}_{A,B}(X)\to {{\mathcal H}}_{A,B}(X)\circ V_A\},$$ where, for any $X\in {{\mathcal B}}(A,B)$, $\sigma_X$ is a natural isomorphism 2-cell such that $$\label{relativ-cent1} \sigma_{I_A}={\mbox{\rm id\,}}_{V_A},
(\alpha_{X,Y}\circ{\mbox{\rm id\,}}_{V_A}) \sigma_{X\circ Y}= ({\mbox{\rm id\,}}_{{{\mathcal H}}(X)}\circ \sigma_Y)(\sigma_X\circ {\mbox{\rm id\,}}_{{{\mathcal H}}(Y)})({\mbox{\rm id\,}}_{V_B}\circ \alpha_{X,Y}),$$ for any 0-cells $A,B, C\in {{\mathcal B}}$, and any pair of 1-cells $X\in {{\mathcal B}}(C,B)$, $Y\in {{\mathcal B}}(A,C)$.

If $(V,\sigma)$, $(W,\tau)$ are two objects in ${{\mathcal Z}}({{\mathcal H}})$, a morphism $f:(V,\sigma)\to (W,\tau)$ in ${{\mathcal Z}}({{\mathcal H}})$ is a collection of 2-cells $f_A:V_A\Rightarrow W_A$, $A\in{{\mathcal B}}$ such that $$\label{relativ-cent2} 
({\mbox{\rm id\,}}_{{{\mathcal H}}(X)}\circ f_A)\sigma_X=\tau_X (f_B\circ {\mbox{\rm id\,}}_{{{\mathcal H}}(X)}),$$ for any 1-cell $X\in{{\mathcal B}}(A,B)$. The category ${{\mathcal Z}}({{\mathcal H}})$ has a monoidal product defined as follows. Let $(V,\sigma), (W,\tau)\in {{\mathcal Z}}({{\mathcal H}})$ be two objects. Then $(V,\sigma){{\otimes}}(W,\tau)=(V{{\otimes}}W, \sigma{{\otimes}}\tau)$, where for 