Why weights are needed once the base of enrichment is not $\mathbf{S}\mathbf{e}\mathbf{t}$

Weighting a limit looks at first like an optional generality: by Weighting by the constant singleton gives exactly the ordinary limit the ordinary limit is the weighted limit at one particular weight, so on this page nothing is lost by working with cones alone. This remark records what that proof spends, and hence why the generality is not optional once the hom-objects of a category are no longer sets.

Two features of $\mathbf{S}\mathbf{e}\mathbf{t}$, and only those two.

First, that a cone is a natural transformation out of a constant weight: the weight $\mathrm{\Delta }\{\ast \}$ exists because $\mathbf{S}\mathbf{e}\mathbf{t}$ has a one-element set and because the assignment sending every object of the index category to it and every morphism to the identity is a functor. Second, that a function out of a one-element set is the same thing as an element of the codomain, which is what turns the components ${\alpha }_j:\{\ast \}\to \mathcal{M}(m,Fj)$ into the legs ${\lambda }_j:m\to Fj$ of a cone.

Neither feature is about limits. Both are statements about the category in which the weight takes its values, which on this page is $\mathbf{S}\mathbf{e}\mathbf{t}$ throughout, by Set-weighted limits and colimits.

Replace the values of the weight by the objects of some other category $\mathcal{V}$, so that a hom-object $\mathcal{M}(m,Fj)$ is an object of $\mathcal{V}$ rather than a set. Then the second feature is no longer available in general: there need not be a canonical Set-like identification between elements and morphisms from the monoidal unit. In bases such as $\mathbf{A}\mathbf{b}$, maps from $\mathbb{Z}$ do recover elements, but that is additional structure of the chosen base rather than a formal property of enrichment. The legs of a cone must therefore be formulated through morphisms of $\mathcal{V}$ into the hom-objects. The first feature is not automatic either, since a constant assignment into $\mathcal{V}$ has to be shown to be a $\mathcal{V}$-functor before it can serve as a weight, and that is a condition on $\mathcal{V}$, not a triviality.

What survives untouched is the definition used on this page: a weighted limit is a representing object for the functor sending $m$ to the transformations out of the weight, and that definition asks nothing of the values of the weight beyond being able to form those transformations. This is the reason the weighted notion, and not the conical one, is the definition that is generalised; Kelly's §3.9 works out the resulting comparison map for a general base and shows what it fails to be.

The development that carries this out is planned for the page enriched-categories and is not available at this point in the reading order, so nothing about enriched limits is asserted here. What is asserted is only what the proofs on this page actually spend, which A weighted limit is an ordinary limit over the category of elements of the weight, and a weighted colimit an ordinary colimit over it makes explicit: the index category is replaced by the category of elements of the weight, and elements are exactly what a general base does not supply.