= Solution
For each $d\in\mathcal D$, let $P(d)=\pi_0(d\downarrow F)$, the collection of connected components, where connectedness means connection by a zigzag of arrows. Precomposition with $u:d\to d'$ gives a function $P(u):P(d')\to P(d)$. Identity and composition are inherited from precomposition.
Construct a <category> $\mathcal E$ whose objects are pairs $(d,c)$ with $c\in P(d)$. An arrow $(d,c)\to(d',c')$ is an arrow $u:d\to d'$ satisfying $c=P(u)c'$. Composition is that of $\mathcal D$, and the component equation is preserved by composition. The projection $p:\mathcal E\to\mathcal D$ is a <discrete fibration>: an arrow $u:d\to d'$ has precisely one lift into $(d',c')$, namely the arrow from $(d,P(u)c')$ represented by $u$.
Define $J:\mathcal C\to\mathcal E$ by
$$
J(a)=(Fa,[(a,1_{Fa})]),\qquad J(j)=F(j).
$$
This is well defined because $(a,1_{Fa})$ and $(a',Fj)$ lie in the same component of $(Fa\downarrow F)$ whenever $j:a\to a'$. Clearly $pJ=F$.
For an object $(d,c)$ of $\mathcal E$, an object of $((d,c)\downarrow J)$ is exactly a pair $(a,u:d\to Fa)$ representing the component $c$. Its morphisms are exactly the arrows in $(d\downarrow F)$ between such objects. Thus this <comma category> is the component $c$ itself, which is nonempty and connected. This proves that $J$ is a <final functor>, and hence
$$
\boxed{F=pJ,\qquad J\text{ final},\quad p\text{ a discrete fibration}.}
$$
This is the <final-discrete-fibration factorization>. For small <categories> the collections $P(d)$ are sets and this construction is the <category of elements> of the presheaf $P$. For <categories> of a larger size, the construction is interpreted in the same chosen ambient size framework.
Back to article page
