= 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 $\bar\lambda_j=\lambda_iD(u)$. Connectedness makes this independent of the choice. Consequently
$$
\operatorname*{colim}_{i\in\mathcal I}D(Fi)\cong
\operatorname*{colim}_{j\in\mathcal J}D(j)
$$
whenever either side is constructed by this universal property.
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