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Every functor factors as a final functor followed by a discrete fibration. For $F:\mathcal{C}\to \mathcal{D}$, the intermediate category has objects $(B,c)$, where $B\in \mathcal{D}$ and $c$ is a connected component of $(B\downarrow F)$. An arrow $(B,c)\to (B^{\prime },c^{\prime })$ is an arrow $g:B\to B^{\prime }$ whose precomposition functor sends $c^{\prime }$ to $c$.

The finality property of a functor means every required comma category is nonempty and connected. It is the condition defining a final functor; the displayed criterion uses categorical connected components.

Given a commutative square whose left functor is final and whose right functor is a discrete fibration, there is a unique diagonal functor filling the square. To construct its value on an object, choose an object of the relevant connected comma category and lift the resulting arrow; uniqueness of lifts makes the answer constant over that connected category.

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

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 $u:d\to d^{\prime }$ gives a function $P(u):P(d^{\prime })\to P(d)$. 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 $(d,c)\to (d^{\prime },c^{\prime })$ is an arrow $u:d\to d^{\prime }$ satisfying $c=P(u)c^{\prime }$. 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 $u:d\to d^{\prime }$ has precisely one lift into $(d^{\prime },c^{\prime })$, namely the arrow from $(d,P(u)c^{\prime })$ represented by $u$.

Define $J:\mathcal{C}\to \mathcal{E}$ byThis is well defined because $(a,1_{Fa})$ and $(a^{\prime },Fj)$ lie in the same component of $(Fa\downarrow F)$ whenever $j:a\to a^{\prime }$. 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 henceThis 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.

Let $F:\mathcal{C}\to \mathcal{D}$ be a final functor, and let ${\lambda }_c:GFc\to V$ be a cocone under a diagram $GF$. For $d\in \mathcal{D}$, choose an object $(c,a:d\to Fc)$ of the nonempty comma category $(d\downarrow F)$ and defineIf $t:(c,a)\to (c^{\prime },a^{\prime })$ is a morphism in that comma category, then $Fta=a^{\prime }$, so the cocone equation gives ${\lambda }_{c}G(a)={\lambda }_{c^{\prime }}G(Ft)G(a)={\lambda }_{c^{\prime }}G(a^{\prime })$. Equality also holds along a reversed arrow, and hence along any zigzag. Connectedness makes the definition independent of the chosen object. For $v:d\to d^{\prime }$, use $(c,a:d^{\prime }\to Fc)$ and $(c,av:d\to Fc)$ to obtain ${\bar{\lambda }}_d={\bar{\lambda }}_{d^{\prime }}Gv$. Thus $\bar{\lambda }$ is a cocone under $G$. Taking $(c,1_{Fc})$ shows that it extends $\lambda$.

Every extension must satisfy ${\bar{\lambda }}_d={\lambda }_{c}G(a)$ by its cocone equation, proving uniqueness. The construction commutes with postcomposition by a morphism $V\to V^{\prime }$. Hence restriction gives a natural bijection between the sets of cocones of $G$ and $GF$ with any fixed vertex. A universal cocone for $GF$ therefore extends to a universal cocone for $G$. In particular,whenever the latter exists, and existence of all colimits of shape $\mathcal{C}$ implies existence of all colimits of shape $\mathcal{D}$.

For the factorization of an arbitrary functor $H:\mathcal{C}\to \mathcal{E}$, letthe set of connected components of a category, where components are defined by finite zigzags. Precomposition by $v:e\to e^{\prime }$ gives a functor $(e^{\prime }\downarrow H)\to (e\downarrow H)$ and therefore a map $P(v):P(e^{\prime })\to P(e)$. Identity and composition laws make $P$ a categorical presheaf. This is a set-valued construction when the relevant comma categories are small or have only a set of components, as in the usual small-category convention for this factorization.

Take $\mathcal{D}$ to be the category of elements of this presheaf. Its objects are pairs $(e,S)$ with $S\in P(e)$; a morphism $(e,S)\to (e^{\prime },S^{\prime })$ is an arrow $v:e\to e^{\prime }$ satisfying $P(v)(S^{\prime })=S$. Define $G:\mathcal{D}\to \mathcal{E}$ by forgetting $S$. Given $v:e\to e^{\prime }$ and an object $(e^{\prime },S^{\prime })$, the unique lift into that object has source $(e,P(v)(S^{\prime }))$ and underlying arrow $v$. This proves that $G$ is a discrete fibration.

Define $F(c)=(Hc,[(c,1_{Hc})])$. For $t:c\to c^{\prime }$, take $F(t)=Ht$. The arrow $t$ in $(Hc\downarrow H)$ joins $(c,1_{Hc})$ to $(c^{\prime },Ht)$, so the required component equation holds. It follows that $F$ is a functor and $GF=H$.

Finally, an object of $((e,S)\downarrow F)$ is a pair $(c,v:e\to Hc)$ whose component in $(e\downarrow H)$ is $S$. Its morphisms are exactly the morphisms of that comma category between objects in $S$. Consequently $((e,S)\downarrow F)$ is precisely the full connected component $S$, and is nonempty and connected. Thus $F$ is final, and every such functor factors as a final functor followed by a discrete fibration.

Let ${\lambda }_A:GF(A)\to Z$ be the given cocone under a diagram. For $B\in \mathcal{D}$, choose an object $(A,u:B\to FA)$ of the comma category $(B\downarrow F)$ and defineSuch a choice exists because $F$ is a final functor. If $a:A\to A^{\prime }$ gives a morphism $(A,u)\to (A^{\prime },u^{\prime })$, so $F(a)u=u^{\prime }$, the cocone relation givesThe connected category condition makes the result independent of the choice along any zigzag.

For $v:B\to B^{\prime }$, choose $(A,u:B^{\prime }\to FA)$; then $(A,uv)$ is available for $B$, giving ${\bar{\lambda }}_B={\bar{\lambda }}_{B^{\prime }}G(v)$. Thus the maps form a cocone under a diagram on $G$. Choosing $(A,1_{FA})$ shows that restriction recovers ${\lambda }_A$. Any extension must satisfy ${\bar{\lambda }}_B={\bar{\lambda }}_{FA}G(u)={\lambda }_{A}G(u)$, proving uniqueness. This is cocone extension along a final functor.

If $GF$ has a colimit, extend its universal cocone in this way. Every cocone on $G$ restricts to one on $GF$, factors uniquely through that colimit, and then agrees with its extension by uniqueness. Hence the same vertex is a colimit of $G$. In particular,and existence of all $\mathcal{C}$-shaped colimits in $\mathcal{E}$ implies existence of all $\mathcal{D}$-shaped colimits.

A cocone under a diagram $D:\mathcal{J}\to \mathcal{C}$ with vertex $X$ is a natural family ${\lambda }_j:Dj\to X$, so ${\lambda }_{k}D(u)={\lambda }_j$ for every $u:j\to k$. A colimit of $D$ is an initial such cocone: for every cocone $\lambda :D\Rightarrow \Delta X$, there is a unique $h:\mathrm{colim}D\to X$ with $h{\iota }_j={\lambda }_j$ for all $j$.

Let $F:\mathcal{I}\to \mathcal{J}$ be a final functor and let ${\lambda }_i:D(Fi)\to X$ be a cocone under $DF$. For each $j$, choose an object $(i,u:j\to Fi)$ of the nonempty comma category $(j\downarrow F)$ and defineA morphism $(i,u)\to (i^{\prime },u^{\prime })$ in the comma category gives $F(v)u=u^{\prime }$, and the cocone identity shows that the two resulting maps are equal. Because $(j\downarrow F)$ is a connected category, a zigzag proves independence of the chosen object. The same construction applied after a morphism $j\to k$ proves naturality, so $\bar{\lambda }$ is a cocone. Any extension must have this value because it must satisfy the cocone identity along $u$, proving uniqueness. This is cocone extension along a final functor.

If $L$ is a colimit of $DF$, its universal cocone extends uniquely to $D$. Restriction and extension give mutually inverse correspondences between cocones from $D$ and from $DF$, so the extended cocone is a colimit of $D$. Therefore the existence of all colimits of shape $\mathcal{I}$ implies the required colimits of shape $\mathcal{J}$.

Now suppose $\mathcal{J}$ is a sifted category. For $D,E:\mathcal{J}\to \mathbf{Set}$, the product functor $-\times A$ is a left adjoint, so it preserves colimits. Applying this once in each variable givesThe diagonal $\Delta :\mathcal{J}\to \mathcal{J}\times \mathcal{J}$ is final, so the right side isThus ${\mathrm{colim}}_{\mathcal{J}}$ preserves binary products in a category. Since $\mathcal{J}$ is connected, the colimit of the constant singleton diagram is a singleton, so it also preserves the terminal object. It therefore preserves finite products.

Consider the commutative squarewhere $F$ is a final functor and $G$ is a discrete fibration. For $B\in \mathcal{B}$, choose an object $(A,\beta :B\to FA)$ of the nonempty comma category $(B\downarrow F)$. Commutativity gives an arrowLift it uniquely through $G$ with codomain $HA$, and define $LB$ to be the domain of this lift.

This definition does not depend on the choice of $(A,\beta )$. A morphism $u:(A,\beta )\to (A^{\prime },{\beta }^{\prime })$ in the comma category satisfies $Fu\beta ={\beta }^{\prime }$. The composite of the lift of $K\beta$ with $Hu$ is then a lift of $K{\beta }^{\prime }$ with codomain $H{A}^{\prime }$, so uniqueness of discrete-fibration lifts says that it is the chosen lift and has the same domain. Since $(B\downarrow F)$ is connected, all choices give the same object $LB$.

For $v:B\to B^{\prime }$, define $Lv$ as the unique lift through $G$ of $Kv:KB\to K{B}^{\prime }=G(L{B}^{\prime })$ with codomain $L{B}^{\prime }$. Its domain is $LB$: choose ${\beta }^{\prime }:B^{\prime }\to FA$, and observe that composing this lift with the lift of $K{\beta }^{\prime }$ yields the lift associated with ${\beta }^{\prime }v:B\to FA$. Uniqueness of lifting also proves preservation of identities and composition, so $L$ is a functor and $GL=K$.

For $B=FA$, choose $(A,1_{FA})$ in $(FA\downarrow F)$. The lift of $1_{GHA}$ is $1_{HA}$, hence $LFA=HA$; the same lifting argument on arrows gives $LF=H$. Finally, if $L^{\prime }$ is another filler, then for every $\beta :B\to FA$, the arrow $L^{\prime }\beta :L^{\prime }B\to HA$ is a lift of $K\beta$. Unique lifting forces $L^{\prime }B=LB$ and then forces equality on arrows. Thus $L$ is unique, proving orthogonality of final functors and discrete fibrations.

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.

A small category $\mathcal{J}$ is sifted when it is connected and its diagonal functor $\mathcal{J}\to \mathcal{J}\times \mathcal{J}$ is a final functor. Equivalently, colimits of shape $\mathcal{J}$ in the Category of sets preserve finite products.