Cocone extension along a final functor

Let $F:\mathcal{I}\to \mathcal{J}$ be final and $D:\mathcal{J}\to \mathcal{C}$. A cocone ${\lambda }_i:D(Fi)\to X$ extends uniquely to $D$: for $j\in \mathcal{J}$, choose $(i,u:j\to Fi)$ in $(j\downarrow F)$ and set ${\bar{\lambda }}_j={\lambda }_{i}D(u)$. Connectedness makes this independent of the choice. Consequentlywhenever either side is constructed by this universal property.