Powers and copowers

Let $\calA$ be a category, enriched in a closed category $\calV$.
($\calV$ closed) means, in particular, that (for all $\objX, \objY \in \calV$ there is an internal hom $\boxed{[\objX,\objY]} \in \calV$, satisfying certain axioms).
($\calA$ enriched in $\calV$) implies that (for all $\objA, \objB \in \calA$ there is a hom object $\boxed{\hom \objA \calA \objB} \in \calV$, satisfying certain axioms).
To say that ($\calA$ admits powers and copowers relative to $\calV$) means precisely that,
(for all $\objX\in\calV$ and $\objA,\objB\in\calA$ there are isomorphisms in $\calV$ as shown below, $\calV$-natural in all their variables).
Shown below the isomorphisms are (the units and counits associated with those natural isomorphisms)
(the identifier $\Eval$ is overloaded, relying on context to disambiguate occurrences of it). \[\begin{array}{cclcc} \hom \objA \calA {\boxed{\scriptstyle \objX\power\objB}} & \buildrel {\textstyle \overline{()}} \over \cong & \big[ \objX, \hom \objA \calA \objB \big] & \buildrel {\textstyle \widetilde{()}} \over \cong & \hom {\boxed{\scriptstyle \objX\copower\objA}} \calA \objB \\ \\ \objA \xrightarrow [ \smash {\textstyle \Eval } ]{} \hom \objA \calA \objB \power \objB && \objX \xrightarrow{\textstyle \pi} \smash { \hom {\objX\power\objB} \calA \objB } \\ && \mkern4mu \Vert \\ && \objX \xrightarrow [ \textstyle \iota ]{} \hom \objA \calA {\objX\copower\objA} && \hom \objA \calA \objB \copower \objA \xrightarrow [ \smash {\textstyle \Eval } ]{} \objB \\ \end{array}\]

---

The following display shows how (an $\objX \xrightarrow{\textstyle \boxed\functionf} \hom \objA \calA \objB$ in $\calV_0$) determines ($\objA \xrightarrow{\textstyle \boxed{\bar\functionf}} \objX\power\objB$ and $\objX\copower\objA \xrightarrow{\textstyle \boxed{\tilde\functionf}} \objB$ in $\calA_0$), and vice versa,
using the units and counits of the isomorphisms (whatever they may be named). \[\begin{array}{} & \calA_0 & & \mkern4em & && \calV_0 && & \mkern4em & & \calA_0 \\ \\ &&&&&& \objX \\ &&&&& {}\rlap{\mkern-.5em \raise4pt\hbox{$\boxed\pi$}} \swarrow && \searrow \rlap{\mkern-1ex \raise4pt\hbox{$\boxed\iota$}} \\ \objX\power\objB && && \hom {\objX\power\objB} \calA \objB && \Bigg\downarrow \rlap{\mkern-13mu\boxed\functionf} && \hom \objA \calA {\objX\copower\objA} &&& & \objX\copower\objA \\ \llap{\functionf\power\objB} \Bigg\uparrow & \nwarrow \rlap{\mkern-1ex \raise4pt\hbox{$\boxed{\bar\functionf}$}} & && & \llap{\hom {\bar\functionf} \calA \functionB} \searrow && \swarrow \rlap{\hom \objA \calA {\tilde\functionf}} & && & {}\rlap{\mkern-.5em \raise4pt\hbox{$\boxed{\tilde\functionf}$}} \swarrow & \Bigg\downarrow \rlap{\functionf\copower\objA} \\ {\hom \objA \calA \objB} \power \objB & \xleftarrow[\textstyle \boxed\Eval]{} & \objA && && \hom \objA \calA \objB && && \objB & \xleftarrow[\textstyle \boxed\Eval]{} & {\hom \objA \calA \objB} \copower \objA \\ \end{array}\]