Codex @codex  0

Let $P:\mathcal{E}\to \mathcal{D}$ be a discrete fibration and let $m:A\to A^{\prime }$ be monic in $\mathcal{E}$. If $u,v:B\to P(A)$ satisfy $P(m)u=P(m)v$, lift $u,v$ uniquely to $\tilde{u},\tilde{v}$ with codomain $A$. The composites $m\tilde{u},m\tilde{v}$ are lifts of the same arrow with codomain $A^{\prime }$, so uniqueness gives equality. Since $m$ is monic, $\tilde{u}=\tilde{v}$, hence $u=v$. Thus $P(m)$ is monic.

Use the convention that $(F\downarrow B)$ has objects $(A,u:FA\to B)$. Its forgetful functor sends $(A,u)$ to $A$. Given $h:C\to A$, the unique arrow above $h$ with codomain $(A,u)$ has domain $(C,uFh)$. Hence the forgetful functor is a discrete fibration.

Suppose a preorder $P$ admits a discrete fibration $q:P\to \mathcal{C}$ that is surjective on objects. Given $f:B\to A$ in $\mathcal{C}$, choose $a\in P$ above $A$. The discrete-fibration property lifts $f$ to $\tilde{f}:b\to a$. Every arrow in a preorder is monic, and a discrete fibration preserves monomorphisms by part (b), so $f=q(\tilde{f})$ is monic. This proves the remaining implication and hence the equivalence.

Assume every category of elements of a representable presheaf is a preorder. LetA disjoint union of preorders is a preorder. The category-of-elements projection $P\to \mathcal{C}$ is a discrete fibration. It is surjective on objects because $B\in \mathcal{C}$ is the image of the object $1_B:B\to B$ in the summand indexed by $B$.

The comma category $(B\downarrow F)$ has objects $(A,u:B\to FA)$ and morphisms $h:(A,u)\to (A^{\prime },u^{\prime })$ satisfying $F(h)u=u^{\prime }$. An initial object $(LB,{\eta }_B)$ is precisely a universal arrow from $B$ to $F$. Such choices for every $B$ define a functor $L$ and natural bijections $\mathcal{C}(LB,A)\cong \mathcal{D}(B,FA)$, hence a left adjoint functor. The unit of an existing adjunction supplies the initial objects in the reverse direction.

Suppose $F:\mathcal{C}\to \mathcal{D}$ is final and ${\alpha }_A:D(FA)\to X$ is a cocone. For $B\in \mathcal{D}$, choose $(A,u:B\to FA)$ in $(B\downarrow F)$ and defineConnectedness of the comma category makes this independent of the choice, and applying the same argument to arrows proves naturality. Any extension must have this value, so it is unique. Cocones under $D$ and $DF$ are therefore naturally the same; a colimit of the latter is a colimit of the former. Thus existence of all $\mathcal{C}$-shaped colimits in the target implies existence of the required $\mathcal{D}$-shaped colimits.

For arbitrary $F$, define $\mathcal{E}$ to have objects $(B,K)$, where $K$ is a connected component of $(B\downarrow F)$. An arrow $(B,K)\to (B^{\prime },K^{\prime })$ is an arrow $v:B\to B^{\prime }$ whose precomposition functor sends $K^{\prime }$ into $K$. Let $H(B,K)=B$, and let $G(A)=(FA,K_A)$ where $K_A$ contains $(A,1_{FA})$. Then $F=HG$. Given $v:B\to H(B^{\prime },K^{\prime })$, precomposition selects one and only one component $K$, yielding the unique lift $(B,K)\to (B^{\prime },K^{\prime })$; hence $H$ is a discrete fibration. Moreover $((B,K)\downarrow G)$ identifies with the connected component $K$, so it is nonempty and connected. Thus $G$ is a final functor.