$ \require{AMScd} \def\C{\mathcal{C}} \def\D{\mathcal{D}} $ Is there such a thing as enriched dinaturality?
I'd take this as the definition:
Let $P,Q : \C°\otimes \C \to \D$ be two $\cal V$-functors; a $\cal V$-dinatural transformation $\alpha : P \overset{..}\Rightarrow Q$ consists of a family of arrows in $\cal V$, $$ \alpha_C : I \to \D(P(C,C), Q(C,C)), $$ indexed by the objects of $\C$, such that for every $C,C'\in\C$ the following commutativity holds: the arrow $$ \begin{CD} \C(C,C')\otimes \C(C,C')\\ @|\\ \C(C,C') \otimes I \otimes \C(C,C') \\ @VV{P(\_,C)\otimes\alpha_C\otimes Q(C,\_)}V\\ \D(P(C',C), P(C,C)) \otimes \D(P(C,C), Q(C,C)) \otimes \D(Q(C,C), Q(C,C'))\\ @VVcV\\ \D(P(C',C), Q(C,C'))\\ \end{CD} $$ equals $$ \begin{CD} \C(C,C')\otimes \C(C,C')\\ @|\\ \C(C,C') \otimes I \otimes \C(C,C') \\ @VV{Q(\_,C')\otimes \alpha_{C'}\otimes P(C',\_)}V\\ \D(Q(C',C'), Q(C,C'))\otimes \D(P(C',C'), Q(C',C'))\otimes \D(P(C',C), P(C',C'))\\ @VVcV\\ \D(P(C',C), Q(C,C'))\\ \end{CD} $$ where I took into account the action of $P,Q$ on arrows as enriched functors, and $c$ is a suitable composition map.
Is this definition correct, and does it capture the correct notion, even if more general than extranaturality? (e.g., does it allow to define co/wedges, co/ends..?)
Edit: It seems that this definition is slightly incorrect, or useless, in that there is no way to impose the commutativity above on the same morphism $f : C \to C'$. This is due to the fact that in general there is diagonal arrow $V \to V\otimes V$ in $\cal V$ that allows to impose that $$ P(f,C)\circ \alpha_C \circ Q(C,f) = Q(f,C')\circ \alpha_{C'}\circ P(C',f) $$ the only condition that the above commutativity imposes is that for a pair of parallel $f,f' : C \to C'$ one has $$ P(f,C)\circ \alpha_C \circ Q(C,f) = Q(f',C')\circ \alpha_{C'}\circ P(C',f') $$ and this makes no sense. I find intriguing that the absence of a "cloning" operation in the monoidal structure prevents dinaturality from existing. Can you tell me why is it so?