A category satisfying the explicit SAFT intersection hypotheses is cocomplete

The diagonal $\mathrm{\Delta }$ preserves all small limits. Let $E:\mathcal{K}\to \mathcal{C}$ be a small diagram with limiting cone $(\lim E,{\lambda }_k)$ in $\mathcal{C}$, which exists by the completeness in [L2]. A cone over $\mathrm{\Delta }E$ with apex $G\in {\mathcal{C}}^{\mathcal{J}}$ is a family of maps $G\to \mathrm{\Delta }E(k)$ natural in $\mathcal{J}$ and compatible over $\mathcal{K}$, so at each $j\in \mathcal{J}$ its components form a cone over $E$ with apex $G(j)$; the universal property of $\lim E$ gives a unique map $G(j)\to \lim E$ for each $j$, and uniqueness makes that family automatically natural in $\mathcal{J}$. Hence $(\mathrm{\Delta }(\lim E),\mathrm{\Delta }{\lambda }_k)$ is a limiting cone and $\mathrm{\Delta }$ is continuous, which is the hypothesis both branches of [L4] require. No selection is involved, because each component mediator is unique. Its domain has the stated SAFT data and its codomain is locally small by [L1], so [L4] gives an initial object in $(D\downarrow \mathrm{\Delta })$ for every $D\in {\mathcal{C}}^{\mathcal{J}}$, including the empty diagram.

An object of $(D\downarrow \mathrm{\Delta })$ is a natural transformation $D\Rightarrow \mathrm{\Delta }X$, that is, a family ${\xi }_j:D(j)\to X$ commuting with the arrows of $\mathcal{J}$ — exactly a cocone under $D$ with vertex $X$ — and its morphisms are the maps of vertices commuting with those families, exactly the morphisms of $\mathrm{Cocone}(D)$. So the initial object of step 1.1 is an initial cocone, which by [L3] is a colimit of $D$. Since $\mathcal{J}$ and $D$ were arbitrary, [L2] makes $\mathcal{C}$ cocomplete. When the initial objects are supplied as a family, the functor form in [L4] identifies their assembly as the colimit functor.