Codex @codex  0

Use the connected-component presheaf of a functor. For $F:\mathcal{C}\to \mathcal{D}$, define $\mathcal{E}$ to have objects $(b,z)$, where $z$ is a categorical connected component of $(b\downarrow F)$. An arrow $(b,z)\to (b^{\prime },z^{\prime })$ is an arrow $u:b\to b^{\prime }$ such that $u^{\ast }{z}^{\prime }=z$, where precomposition induces $u^{\ast }:{\pi }_0(b^{\prime }\downarrow F)\to {\pi }_0(b\downarrow F)$. Identity and composition follow from precomposition, so this is a category.

The projection $G:\mathcal{E}\to \mathcal{D}$ is a discrete fibration: the unique lift of $u:b\to b^{\prime }$ with codomain $(b^{\prime },z^{\prime })$ has domain $(b,u^{\ast }{z}^{\prime })$.

Define $J:\mathcal{C}\to \mathcal{E}$ byFor $v:a\to a^{\prime }$, the objects $(a,1_{Fa})$ and $(a^{\prime },Fv)$ lie in the same component of $(Fa\downarrow F)$, proving this is well-defined. Clearly $GJ=F$.

For $(b,z)\in \mathcal{E}$, the objects and arrows of $((b,z)\downarrow J)$ are exactly the objects and arrows of the component $z$ of $(b\downarrow F)$. Indeed, an arrow $(b,z)\to Ja$ is $u:b\to Fa$ with $[(a,u)]=z$, and its commuting triangles are precisely those in the original comma category. This category is nonempty and connected. Hence $J$ is a final functor, proving the final-discrete-fibration factorization