Set-weighted limits and colimits

Let $\mathcal{J}$ be a small category, let $\mathcal{M}$ be locally small (Small, locally small, and large categories) and let $D:\mathcal{J}\to \mathcal{M}$ be a diagram. Because $\mathcal{J}$ is small and $\mathbf{S}\mathbf{e}\mathbf{t}$ is locally small, the functor category $[\mathcal{J},\mathbf{S}\mathbf{e}\mathbf{t}]$ is locally small (Functor category $[\mathcal{C},\mathcal{D}]$, If $\mathcal{C}$ is small and $\mathcal{D}$ is locally small then $[\mathcal{C},\mathcal{D}]$ is locally small; if both are small it is small, Sets and functions form the large locally small category $\mathbf{S}\mathbf{e}\mathbf{t}$), so each collection of natural transformations named below is a set (Natural transformation and its components).

A weight for a limit is a functor $W:\mathcal{J}\to \mathbf{S}\mathbf{e}\mathbf{t}$. A weighted limit $\{W,D\}$ is an object of $\mathcal{M}$ that represents the functor sending an object to the set of natural transformations from the weight (Presheaves, covariantly and contravariantly representable functors, and representations), namely

$\mathcal{M}(-,\{W,D\})\cong [\mathcal{J},\mathbf{S}\mathbf{e}\mathbf{t}](W,\mathcal{M}(-,D-)):{\mathcal{M}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t},$

where $\mathcal{M}(m,D-):\mathcal{J}\to \mathbf{S}\mathbf{e}\mathbf{t}$ is the covariant hom-functor $\mathcal{M}(m,-)$ composed with $D$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category). Written out, the isomorphism is a bijection natural in $m$ between morphisms $m\to \{W,D\}$ and families of functions ${\alpha }_j:W(j)\to \mathcal{M}(m,Dj)$ satisfying $D(u)\circ {\alpha }_j(w)={\alpha }_k(W(u)(w))$ for every $u:j\to k$ and $w\in W(j)$.

A weight for a colimit is a presheaf $W:{\mathcal{J}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$). A weighted colimit $W\star D$ is an object of $\mathcal{M}$ with

$\mathcal{M}(W\star D,-)\cong [{\mathcal{J}}^{\mathrm{o}\mathrm{p}},\mathbf{S}\mathbf{e}\mathbf{t}](W,\mathcal{M}(D-,-)):\mathcal{M}\to \mathbf{S}\mathbf{e}\mathbf{t},$

naturally in the second variable, where $\mathcal{M}(D-,m):{\mathcal{J}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ sends $j$ to $\mathcal{M}(Dj,m)$ and is obtained by composing the contravariant hom-functor $\mathcal{M}(-,m):{\mathcal{M}}^{\mathrm{o}\mathrm{p}}\to \mathbf{S}\mathbf{e}\mathbf{t}$ with $D^{\mathrm{o}\mathrm{p}}:{\mathcal{J}}^{\mathrm{o}\mathrm{p}}\to {\mathcal{M}}^{\mathrm{o}\mathrm{p}}$.

The natural transformation corresponding to the identity of $\{W,D\}$ is the counit cylinder of the weighted limit, with components ${\kappa }_j:W(j)\to \mathcal{M}(\{W,D\},Dj)$; dually for a weighted colimit. Neither object need exist.

The variances are the ones that read correctly against the presheaf convention in force in this library: a weight for a limit is covariant, matching the covariant hom-functor $\mathcal{M}(m,D-)$, and a weight for a colimit is contravariant, matching $\mathcal{M}(D-,m)$. Writing a colimit weight covariantly would ask for a natural transformation between functors of opposite variance, which is not a well-formed condition.

Nothing here mentions cones. A cone over $D$ is recovered by taking the weight that is constantly a one-element set, and that this reproduces the ordinary limit is a theorem rather than a convention: Weighting by the constant singleton gives exactly the ordinary limit.