It's well known that a category $\mathcal{C}$ having (conical) limits/colimits of shape $\mathcal{D}$ is equivalent to the diagonal functor $\Delta^\mathcal{D}\_\mathcal{C}:\mathcal{C}\to\mathcal{C}^\mathcal{D}$ having a left/right adjoint respectively. 



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> Is there a similar characterization of weighted limits/colimits as left/right adjoints, maybe to a diagonal functor into $\mathcal{C}^{(\mathcal{V}^\mathcal{D})}$ for $\mathcal{V}$ a closed symmetric monoidal category or some such? 
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Going through Awodey's introductory category theory text there was no mention of weighted (co)limits, but I need to step up to $2$-category theory for Borceux/Janelidze's (non-Galoisian) categorical Galois theory and weighted (co)limits give the correct fully weak notion of $2$-categorical (co)limits.


I've seen it [mentioned](https://mathoverflow.net/questions/97648/weighted-limits-and-completeness) elsewhere that weighted (co)limits are equivalent to conical (co)limits and cotensoring over the walking arrow category, but cotensoring isn't mentioned in Awodey either so this characterization is somewhat opaque to me. The adjoint characterization of (co)limits is very clear, so hopefully an adjoint characterization of weighted (co)limits can demystify their workings as well.


If they aren't special cases of ($1$-)adjoints that would be interesting too, since the moral of $1$-category theory (as I understand it) is that 'every ubiquitous concept in regular mathematics is an adjunction or an adjoint functor'. It is noted in the answer [here](https://math.stackexchange.com/questions/1885386/familiar-categorical-limits-viewed-elegantly-as-weighted-limits) that weighted limits are a fundamentally $2$-categorical notion and considering them on a $1$-categorical level necessarily misses some of the subtleties involved, so perhaps they are a particular case of a $2$-adjunction?



Weighted (co)limits as adjunctions
Here's one way in which weighted limits are adjoints. Let $D$ be a $V$-category and $W:D\to V$ a weight, and $C$ a $V$-category. Suppose that $C$ has copowers, so for $k\in V$ and $x,y\in C$ we have $C(k\odot x,y) \cong V(k,C(x,y))$. Then there is a $V$-functor $\Delta^W:C \to C^D$ defined by $\Delta^W(c)(d) = W(d) \odot c$, and a right adjoint to this functor sends $F\in C^D$ to an object $l$ such that
$$C(c,l) \cong C^D(W(-)\odot c,F) \cong V^D(W,C(c,F))$$
which is the definition of a weighted limit $\lim^W F$.


You can get rid of the assumption of copowers if you take $W$ as the argument of the functor rather than $F$: there's a contravariant functor $\nabla^F : C^{op} \to V^D$ defined by $\nabla^F(c)(d) = C(c,F(d))$, and a mutual right adjoint to this functor sends $W\in V^D$ to an object $l$ such that
$$C(c,l) \cong V^D(W,C(c,F(d)))$$
once again the definition of a weighted limit $\lim^W F$.


