I'm studying the notion of a a $\mathcal{V}$ category $\underline{\mathscr{A}}$ which is powered or compowered over $\mathcal{V}$. I'm having trouble finding a proof that powering/copowering gives a $\mathcal{V}$ functor $\underline{\mathcal{V}} \otimes \underline{\mathscr{A}} \to \underline{\mathscr{A}}$. (Add "op" as necessary.)
In Riehl's Categorical Homotopy Theory, she gives some indication of the proof, but I'm having trouble parsing it. In particular, see these two pages. The claim here appears to be the following:
Proposition: [Edit: I believe I have interpreted the text wrong, because this seems to be incorrect] Let $\underline{\mathscr{A}}$ and $\underline{\mathscr{B}}$ be $\mathcal{V}$ categories, with underlying categories $\mathscr{A}$ and $\mathscr{B}$. Let $L:\mathscr{A} \leftrightarrows \mathscr{B}: R$ be an adjunction with unit $\eta: 1_\mathscr{A} \Rightarrow RL$. Let there be a given isomorphism $\underline{\mathscr{A}}(a,Rb) \cong \underline{\mathscr{B}}(La, b)$ whose image in the underlying categories is the natural isomorphism of the adjunction. Let $R$ be the underlying functor of a $\mathcal{V}$ functor $\mathbf{R}:\underline{\mathscr{B}} \to \underline{\mathscr{A}}$. Then there is a $\mathcal{V}$ functor $\mathbf{F}:\underline{\mathscr{B}} \to \underline{\mathscr{A}}$ agreeing with $F$ on objects and with action on hom objects given by $$\mathbf{F}: \underline{\mathscr{A}}(a,a') \xrightarrow{(\eta_{a'})_*} \underline{\mathscr{A}}(a, RLa') \cong \underline{\mathscr{B}}(Fa, Fa')$$ $\mathbf{F}$ is the $\mathcal{V}$-adjunct of $G$.
I can't seem to show functoriality of $\mathbf{F}$. (See this link where I've drawn the diagrams out.) Maybe someone can help.
It could be that I'm misunderstanding the claim she's making.
Edit: Just considered the possibility that there's something special about $\underline{\mathcal{V}} = \underline{\mathscr{A}}$, which seems to be the case she's dealing with exclusively. Let me see if I can make that work.
Edit 2 The statement in Kelly is
When [the forgetful functor] is conservative--faithfulness is not enough--the existence of a left adjoint $S_0$ for $T_0$ implies that of a left adjoint $S$ for $T$.
So I think I must be misunderstanding Riehl's text here. Perhaps someone can help me.
Let me start by stating clearly the unenriched version of the result whose enriched version I was trying to evoke here. This appears as Proposition 4.3.4 in Category Theory in Context which is available here:
Category Theory in Context
Prop. Consider a functor $G\colon B \to A$ so that for each $a \in A$ there exists an object $Fa \in B$ together with an isomorphism
$B(Fa,b) \cong A(a,Gb)$ that is natural in $b \in B$.
Then there is a unique way to extend the assignment $F \colon obA \to obB$ to a functor $F \colon A \to B$ so that these isomorphisms are also natural in $A$.
The proof is by the Yoneda lemma (see the cited notes).
Now I claim that the same result is true for $V$-categories $A$ and $B$, a $V$-functor $G \colon B \to A$, and a family of isomorphisms $B(Fa,b) \cong A(a,Gb)$ in $V$ that are $V$-natural in $b \in B$.
The given isomorphisms in $V$ can be used to define a canonical map in $V$
$A(a,a') \to V(B(Fa',b), B(Fa,b))$
for each $b \in B$. Not only is this canonical but it's uniquely determined if you want the iso to be $V$-natural in $A$ in the end.
Moreover these maps are (extra-ordinarily) $V$-natural in $B$. What this means is that you actually get a map into the enriched end
$A(a,a') \to \int_{b \in B} V(B(Fa',b), B(Fa,b))$.
(These are defined in section 7.3.) Now one version of the $V$-Yoneda lemma tells you that
$\int_{b \in B} V(B(Fa',b), B(Fa,b)) \cong B(Fa,Fa')$
so this map is the map you seek.
In the definition of tensors I should have asked that the isomorphisms
$M(v\otimes m,n)\cong V(v, M(m,n))$
were $V$-natural in $n$. (Vladimir's comment is correct.) In any case it was absolutely my intention to define tensored to mean that the hom bifunctor has a left $V$-adjoint. Apologies for the confusion!