Equivalent definitions of weighted limits

Question

Consider a symmetric monoidal closed, complete and cocomplete category $\mathcal{V}$. Let $\mathcal{A,C}$ be $\mathcal{V}$-categories and $\mathcal{D}:\mathcal{A} \rightarrow \mathcal{C}$, $\mathcal{W}:\mathcal{A} \rightarrow \mathcal{V}$ be $\mathcal{V}$-functors. The usual definition of a $\mathcal{W}$-weighted limit of $\mathcal{D}$ is as a $\mathcal{V}$-representing object, ie. a $\mathcal{V}$-natural isomorphism in $K$ $$\mathcal{V}(K,\mathcal{W}\text{-}\lim\mathcal{D}) \cong [\mathcal{A,V}](\mathcal{W},\mathcal{C}(K,\mathcal{D}))$$

Note that $\mathcal{V}$ being complete and cocomplete makes it have all weighted limits in this sense.

Now there seems to be another definition of weighted limit, stating that a $\mathcal{V}$-natural transformation $\mathcal{W} \Rightarrow \mathcal{C}(L,\mathcal{D})$ exhibits $L$ as a weighted limit, if for any $K\in\mathcal{C}$ "the induced map" $$\mathcal{C}(K,L) \rightarrow \mathcal{W}\text{-}\lim \mathcal{C}(K,\mathcal{D})$$ is an isomorphism in $\cal{V}$. I am fairly sure this will turn out to be an embarrassingly obvious thing, but I don't see it:

1. How is this "induced map" defined?
2. How do I show that both definitions are equivalent?

I do see that if the weighted limit exists in $\cal{C}$, by continuity of the enriched Yoneda-embedding we will have a chain of natural isomorphisms in $K$ $$\mathcal{W}\text{-}\lim\mathcal{C}(K,\mathcal{D}) \cong \mathcal{C}(K,\mathcal{W}\text{-}\lim \mathcal{D}) \cong [\mathcal{A,V}](\mathcal{W},\mathcal{C}(K,\mathcal{D}))$$ Hence, if we can show that the left map is given by that "induced map" and if there always exists an isomorphism between the first and third term, then I see why (2.) holds true. I didn't manage to find such an isomorphism though.

My best take on (1.) is that we ought to have a chain of isomorphisms $$\begin{align*} &\;\;\;\;\;\mathcal{V}(\mathcal{C}(K,L),\mathcal{W}\lim\mathcal{C}(K,\mathcal{D}))\\ &\cong [\mathcal{A,V}](\mathcal{W},\mathcal{V}(\mathcal{C}(K,L),\mathcal{C}(K,\mathcal{D})))\\ &\cong [\mathcal{A,V}](\mathcal{W},\mathcal{C}(L,\mathcal{D})) \end{align*}$$ where the last isomorphism should be consequence of the fact that the enriched Yoneda-embedding is full and faithful. But being uncomfortable with enriched Yoneda, I am not quite sure if this is correct. If it is, any given $\cal{V}$-natural transformation $\mathcal{W} \Rightarrow \mathcal{C}(L,\mathcal{D})$ gives an induced map...

Thank you very much for your help.

Answer 1

First things first. Let $E : \mathcal{A} \to \mathcal{V}$ be any $\mathcal{V}$-enriched diagram. Then $$\mathcal{V} (T, [\mathcal{A}, \mathcal{V}] (W, E)) \cong [\mathcal{A}, \mathcal{V}] (W, \mathcal{V} (T, E))$$ and this is $\mathcal{V}$-natural in $T$ and $E$ and $W$. Therefore $\mathop{W \textrm{-lim}} E$ exists in $\mathcal{V}$ and is ($\mathcal{V}$-naturally isomorphic to) $[\mathcal{A}, \mathcal{V}] (W, E)$.

Secondly. The $\mathcal{V}$-enriched composition of $\mathcal{C}$ gives a morphism in $\mathcal{V}$, $$\mathcal{C} (L, D a) \otimes \mathcal{C} (K, L) \to \mathcal{C} (K, D a)$$ and this is $\mathcal{V}$-natural in $K$ and $a$. If we have a morphism $W a \to \mathcal{C} (L, D a)$ that is $\mathcal{V}$-natural in $a$ then we can precompose it to obtain $$W a \otimes \mathcal{C} (K, L) \to \mathcal{C} (K, D a)$$ and then use adjunction to get $$\mathcal{C} (K, L) \to [W a, \mathcal{C} (K, D a)]$$ where now we instead have $\mathcal{V}$-extranaturality in $a$, hence a morphism $$\mathcal{C} (K, L) \to \int_{a : \mathcal{A}} [W a, \mathcal{C} (K, D a)]$$ but almost by definition, for any $X : \mathcal{A} \to \mathcal{V}$, $$\int_{a : \mathcal{A}} [W a, X a] \cong [\mathcal{A}, \mathcal{V}] (W, X)$$ and this is $\mathcal{V}$-natural in $X$. Therefore a $\mathcal{V}$-natural transformation $W \Rightarrow \mathcal{C} (L, D)$ induces a morphism $\mathcal{C} (K, L) \to [\mathcal{A}, \mathcal{V}] (W, \mathcal{C} (K, D))$, and this is $\mathcal{V}$-natural in $K$.

Finally, recall that there are really two components in a representation of a functor $F : \mathcal{C}^\textrm{op} \to \mathcal{V}$: a representing object $L$, of course, but also a universal "element" of $F L$ (i.e. a morphism $u : I \to F L$ in $\mathcal{V}$). Universality means that the composite $$\mathcal{C} (K, L) \cong \mathcal{C} (K, L) \otimes I \to \mathcal{C} (K, L) \otimes F L \to F K$$ (where the first isomorphism is the unitor and the third morphism is the adjoint of the morphism $\mathcal{C} (K, L) \to \mathcal{V} (F L, F K)$ in $\mathcal{V}$ given by the action of $F$) is an isomorphism $\mathcal{V}$-natural in $K$. Putting $F K = \mathop{W \textrm{-lim}} \mathcal{C} (K, D) \cong [\mathcal{A}, \mathcal{V}] (W, \mathcal{C} (K, D))$ gives the equivalence of the two definitions you asked about.

(Incidentally, yes, any $W \Rightarrow \mathcal{C} (L, D)$ whatsoever induces a morphism $\mathcal{C} (K, L) \to \mathop{W \textrm{-lim}} \mathcal{C} (L, D)$. This does not depend on $L$ being a $W$-weighted limit of $D$ either. It is universality that is important!)