Enriched weighted limit

Assume $\mathcal{V}$ is symmetric monoidal right closed and its collection of objects is a set, so that it is enriched in itself and enriched opposites are defined using the symmetry (A closed monoidal category is enriched in itself). Let $\mathcal{A}$ be a small $\mathcal{V}$-category, let $T:\mathcal{A}\to \mathcal{B}$ be a $\mathcal{V}$-functor, and let $W:\mathcal{A}\to \mathcal{V}$ be a $\mathcal{V}$-functor (Enriched functor).

An enriched weighted limit of $T$ by $W$ is an object $\{W,T\}$ of $\mathcal{B}$ together with an isomorphism in $\mathcal{V}$

$\mathcal{B}(B,\{W,T\})\cong [\mathcal{A},\mathcal{V}](W,\mathcal{B}(B,T-))$

natural in $B$, whenever the enriched functor category and the displayed hom-object are formed. Dually, for a weight on ${\mathcal{A}}^{\mathrm{o}\mathrm{p}}$, an enriched weighted colimit is an object $W\star T$ with

$\mathcal{B}(W\star T,B)\cong [{\mathcal{A}}^{\mathrm{o}\mathrm{p}},\mathcal{V}](W,\mathcal{B}(T-,B))$

natural in $B$.

This is the direct enriched analogue of Set-weighted limits and colimits: sets are replaced by objects of $\mathcal{V}$, hom-sets by enriched hom-objects, and ordinary natural transformations by enriched ones.