Change of base for enriched categories

The following is a very incomplete preliminary draft

$\begin{array}{lccccccc} \text{The 2-cell} && && \twocellctblr \eta \Phi {\Phi'} \calV {\mkern.5em \calV'} & \mkern6em & \subset & \MONCAT\\ && && \smash{\lower1.6ex\hbox{_}} \\ \text{is mapped by the 2-functor (see CC II.6.3)} \mkern4em && && \Bigg\downarrow \rlap{()_\ast} &&& \smash{\Bigg\downarrow\rlap{()_\ast}} \\ \\ \text{to the 2-cell on the right in} && \catI & \twocellctb \functorT \catA {\catA'} & \twocellctblr {\eta_\ast} {\Phi_\ast} {\Phi_\ast'} {\calV\mathord-\Cat} {\mkern.5em \calV'\mathord-\Cat} && \subset & \CAT \\ \end{array}$

Evaluating the above, we get (the diamond-shaped diagram in $\calV'\mathord-\Cat$ at left below);
evaluating (that diamond-diagram in $\calV'\mathord-\Cat$) at (objects $\objecta,\objecta'$ in $\catA$) (and thus also in $\catA\Phi_\ast$),
we get (the diagram of hom-objects and arrows in $\calV'$ shown at its right): $$\begin{array}{} & {}\rlap{\mkern-2em\calV\mathord-\Cat} & && \calV'\mathord-\Cat && && && && && \calV' \\ \\ && && \catI && && && && && \hom \objecta {(\catA\Phi_\ast)} {\objecta'} \\ && && \llap\objecta \downdownarrows \rlap{\objecta'} && && && && && \Vert \rlap{\text{CC I(6.2)}} \\ && \mkern-3em \catA && \catA\Phi_\ast && \Phi_\ast && && && \llap{\hom \objecta {(\functorT\Phi_\ast)} {\objecta'}} \swarrow && (\hom \objecta \catA {\objecta'}) \phi && \searrow \rlap{ \hom \objecta {(\catA\eta_\ast)} {\objecta'} }\\ & \mkern-1em \llap\functorT \swarrow & & \llap{\functorT\Phi_\ast} \swarrow && \searrow \rlap{\catA\eta_\ast} && \mkern-1.5em \searrow\mkern-.85em\searrow \rlap{\eta_\ast} && && && \llap{(\hom {\objecta} \functorT {\objecta'})\phi} \swarrow && \searrow \rlap{(\hom \objecta \catA {\objecta'})\eta} \\ \catA' && \catA'\Phi_\ast && \functorT\eta_\ast && \catA\Phi_\ast' && \mkern-1.3em \Phi_\ast' & \mkern2em \smash{\mathop\mapsto\limits^{\textstyle \hom \objecta {()} {\objecta'}}} \mkern2em & \hom {\objecta\functorT} {(\catA'\Phi_\ast)} {\objecta'\functorT} & = & (\hom {\objecta\functorT} {\catA'} {\objecta'\functorT}) \phi && (\hom \objecta \functorT {\objecta'}) \eta && (\hom {\objecta} {\catA} {\objecta'}) \phi' & = & \hom {\objecta} {(\catA\Phi_\ast')} {\objecta'} \\ && & \llap{\catA'\eta_\ast} \searrow && \swarrow \rlap{\functorT\Phi_\ast'} && && && && \llap{(\hom {\objecta\functorT} {\catA'} {\objecta'\functorT}) \eta} \searrow && \swarrow \rlap{(\hom {\objecta} {\functorT} {\objecta'}) \phi'} \\ && && \catA'\Phi_\ast' && && && && \llap{\hom \objecta {(\catA'\eta_\ast)} {\objecta'}} \searrow && (\hom {\objecta\functorT} {\catA'} {\objecta'\functorT}) \phi' && \swarrow \rlap{\hom \objecta {(\functorT\Phi_\ast')} {\objecta'}} \\ && && && && && && && \Vert \rlap{\text{CC I(6.2)}} \\ && && && && && && && \hom {\objecta\functorT} {(\catA'\Phi_\ast')} {\objecta'\functorT} \\ \end{array}$$

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References

CC, Eilenberg and Kelly, “Closed Categories”, 1966