 Solution

Consider one connected component $K$ of the category of cocones under a small diagram $D:\mathcal{J}\to \mathbf{Fld}$. Replace every apex by the subfield generated by the images of the fields $D(j)$. These generated fields have cardinality bounded in terms of the small diagram, so they admit a small skeleton, cofinal within $K$.

The apex functor on this small connected category has a limit $L_K$ in $\mathbf{Fld}$ by part (b). For each $j$, the maps from $D(j)$ to all apexes form a compatible cone and therefore induce $D(j)\to L_K$. These maps form a cocone ${\lambda }_K$ under $D$. Its limiting projections give a unique morphism from ${\lambda }_K$ to every cocone in $K$, so ${\lambda }_K$ is initial in that component. Choosing one ${\lambda }_K$ for each component gives a multicolimit.