Conical weights are a proper special case of enriched weights

By [L1], a conical limit uses the constant weight at the tensor unit. By [L2], weighted limits already include the one-object weights given by arbitrary objects $X\in \mathcal{V}$; these are the cotensors $X⋔C$. Thus conical weights constitute only the tensor-unit case of this family.

For $\mathcal{V}=\mathbf{A}\mathbf{b}$, the tensor unit is $\mathbb{Z}$, while $X=\mathbb{Z}\mathrm{/}2$ is a legitimate nonunit weight. Its weighted-limit universal property is $\mathcal{B}(B,(\mathbb{Z}\mathrm{/}2)⋔C)\cong [\mathbb{Z}\mathrm{/}2,\mathcal{B}(B,C)],$ whereas the conical weight on the one-object free enriched category is the constant weight $\mathbb{Z}$. Since $\mathbb{Z}\mathrm{/}2\cong ̸\mathbb{Z}$, these are different weights and different specified universal properties.

Thus the class of enriched weights is strictly larger than the class of conical weights, already over $\mathbf{A}\mathbf{b}$. A particular cotensor may nevertheless be computable from conical limits—for example, when it exists, $(\mathbb{Z}\mathrm{/}2)⋔C$ can be the conical equalizer of $2_C$ and $0_C$. The distinction proved here is between the weighted problems, not a prohibition on alternative constructions of their representing objects.