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A cocomplete abelian category is finitary in the directed-union sense if every cocone under a diagram indexed by a nonempty directed poset, with all its legs monomorphisms, induces a monomorphism from the colimit to the cocone vertex. The category of abelian groups has this property: two colimit representatives mapped to the same vertex become equal at a common upper index because that cocone leg is injective. This use of finitary concerns directed unions, rather than the preservation-of-filtered-colimits meaning used for finitary monads.

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.

First address the unnumbered preliminaries. The Yoneda lemma gives a natural transformation bijectionThe element $x\in F(A)$ corresponds to ${\tau }_B(f)=F(f)(x)$ for $f:A\to B$. These bijections are natural in $A$ and $F$; the contravariant form replaces $\mathcal{C}(A,-)$ by $\mathcal{C}(-,A)$.

Let $E=(1\downarrow F)$ be the category of elements. Its objects are $(A,x)$ with $x\in F(A)$; an arrow $u:(A,x)\to (B,y)$ is $u:A\to B$ with $F(u)(x)=y$. Since $\mathcal{C}$ is a small category, $E$ is small. Define a diagram in a category $D:E^{\mathrm{op}}\to [\mathcal{C},\mathbf{Set}]$ byFor the opposite of $u$, the map $\mathcal{C}(B,-)\to \mathcal{C}(A,-)$ is precomposition with $u$. The Yoneda lemma gives a cocone under a diagram $D\to F$ whose $(A,x)$ component sends $f:A\to B$ to $F(f)(x)$.

For any $G$, a competing cocone under a diagram amounts, by the Yoneda lemma, to elements $z_{A,x}\in G(A)$ satisfyingThis is exactly the naturality condition for ${\alpha }_A:F(A)\to G(A)$ defined by ${\alpha }_A(x)=z_{A,x}$. It gives a unique natural transformation $\alpha :F\to G$ factoring the cocone. Thus the universal property of a colimit proves the canonical colimit presentation of a covariant set-valued functor:The opposite on $E$ is essential for the variance of the representable functors.

We now prove the four conditions equivalent by the cycle $(i)\Rightarrow (ii)\Rightarrow (iii)\Rightarrow (iv)\Rightarrow (i)$. Suppose $F$ is a finite-limit-preserving set-valued functor. For a finite diagram in a category in the comma category $(S\downarrow F)$, write its objects as $(A_i,s_i:S\to F(A_i))$. Take $L={\lim }_i{A}_i$ in $\mathcal{C}$. Since $F$ preserves this finite limit, the compatible maps $s_i$ determine a unique $s:S\to F(L)$ whose composites with $F(L)\to F(A_i)$ are $s_i$. The pair $(L,s)$ has the required categorical limit property in $(S\downarrow F)$, by the same universal properties. For the empty diagram, $F(1)$ is a singleton, so there is exactly one map $S\to F(1)$; this provides the terminal object. Hence $(i)\Rightarrow (ii)$ for every set $S$.

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 reflexive pair $f,g:A\rightrightarrows B$ has a common section $r:B\to A$ with $fr=gr=1_B$. Form the pushout in a category of $f$ and $g$, and write its two maps from $B$ as $j_1,j_2$. Its relation $j_{1}f=j_{2}g$, composed with $r$, gives $j_1=j_2=q$. Thus $qf=qg$. Any $h:B\to C$ with $hf=hg$ gives the compatible pair $(h,h)$ for this pushout and therefore factors uniquely through $q$. This proves the coequalizer of a reflexive pair from a pushout construction.

There is a genuine transcription difference: the original PDF says finite products in the finite-colimit assertion, while the TeX says finite coproducts. The PDF assertion is false with products. Even retaining a pushout-existence assumption, the full subcategory of the Category of sets on nonempty sets has finite products, pushouts, and all coequalizers, but no initial object, hence no empty colimit. Regard the poset with elements $0,a,b,u_0,u_1,\dots$, orderand incomparable $a,b$ as a category. It has finite meets and top $u_0$, hence finite products in a category. Every reflexive pair in a poset is an equal pair and has its identity as a coequalizer. But $a,b$ have no least upper bound, so they have no coproduct in a category.

For the corrected construction using finite coproducts, let $D:\mathcal{J}\to \mathcal{C}$ be a finite diagram in a category. PutDefine $s,t:Y\rightrightarrows X$ on the summand indexed by $a:i\to j$ as ${\nu }_i$ and ${\nu }_{j}D(a)$. Maps $h:X\to Z$ with $hs=ht$ are exactly cocone under a diagram data on $D$. Hence their universal coequalizer is its colimit. Replace $s,t$ by the reflexive pairwhose common section is the second injection and whose coequalizer is unchanged. This proves construction of finite colimits from coproducts and reflexive coequalizers, including the empty diagram via the empty coproduct:

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.