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A cocone under $D:\mathcal{J}\to \mathcal{C}$ with vertex $A$ is a natural family ${\lambda }_j:D(j)\to A$ satisfying ${\lambda }_{k}D(u)={\lambda }_j$ for every $u:j\to k$. Cocones are cones in the opposite category.

In a pointed category, a normal monomorphism is a monomorphism that is the kernel in a category of some morphism. Suppose $m:M\hookrightarrow A$ is the kernel of $f:A\to B$, and let $q:A\to Q$ be the cokernel in a category of $m$. Since $fm=0$, there is a unique $\bar{f}:Q\to B$ with $f=\bar{f}q$. If $qx=0$, then $fx=\bar{f}qx=0$, so the universal property of $m=\ker f$ factors $x$ uniquely through $m$. Therefore $m=\ker q=\ker (\mathrm{coker}m)$. The converse is immediate: if $m$ is the kernel of its cokernel, it is the kernel of a morphism and hence normal.

An abelian category is an additive category with kernels and cokernels in which every monomorphism is normal and every epimorphism is a conormal epimorphism. Finite biproducts and kernels give finite limits. The image and coimage in an abelian category give every $f:A\to B$ the canonical factorizationand the middle map is an isomorphism. The first map is a cokernel and therefore a regular epimorphism. Every epimorphism in an abelian category is the cokernel of its kernel, and epimorphisms are stable under pullback; consequently regular epimorphisms are pullback-stable. This proves that every abelian category is regular.

Define the additive indexing category for chain complexes $\mathbf{Z}$ as follows. Its objects are the integers andLet the generator of $\mathbf{Z}(n,n)$ be $1_n$ and the generator of $\mathbf{Z}(n,n-1)$ be ${\partial }_n$. Composition is bilinear, the $1_n$ are identities, andbecause the target hom-group $\mathbf{Z}(n,n-2)$ is zero. An additive functor $C:\mathbf{Z}\to \mathcal{A}$ chooses objects $C_n=C(n)$ and differentials $d_n=C({\partial }_n)$ satisfying $d_{n-1}{d}_n=0$, hence a complex in an abelian category. Conversely every chain complex defines this unique additive functor.

For self-duality, put $Z_n=\ker d_n$, $B_n=\mathrm{im}{d}_{n+1}$, and let $q:C_n\to Q_n=\mathrm{coker}{d}_{n+1}$. Since $d_n{d}_{n+1}=0$, there is a unique ${\bar{d}}_n:Q_n\to C_{n-1}$ with ${\bar{d}}_{n}q=d_n$. The image-to-kernel factorization gives a canonical isomorphismPassing to the opposite category exchanges kernels with cokernels and images with coimages. The usual construction in ${\mathcal{A}}^{\mathrm{op}}$ is therefore the expression on the right, which is canonically the original homology object. This proves the self-duality of homology.

A diagram in a category is a functor $D:\mathcal{J}\to \mathcal{C}$. A cone over a diagram with vertex $A$ is a familysuch that $D(u){\gamma }_i={\gamma }_j$ for every $u:i\to j$. A categorical limit is a terminal cone: for every cone $(A,\gamma )$ there is a unique map $A\to \lim D$ commuting with all legs.

Suppose $\mathcal{C}$ has small products and equalizers. For a small diagram $D$, formThere are two maps $r,s:P\rightrightarrows Q$. In the coordinate indexed by $u:i\to j$, letThe equalizer $E\to P$ imposes exactly the cone equations. Maps $A\to E$ are therefore naturally the same as cones from $A$ to $D$, so $E\cong \lim D$. This is the construction of small limits from products and equalizers.

Let $F:\mathcal{I}\to \mathcal{J}$ be initial, so every $(F\downarrow j)$ is nonempty and connected. Restriction sends a cone $(A,{\gamma }_j)$ over $D$ to $(A,{\gamma }_{Fi})$ over $DF$. Conversely, given a cone $(A,{\delta }_i)$ over $DF$, choose an objectand defineA morphism in the comma category shows that this expression is unchanged along one edge, and connectedness makes it independent of the chosen object. The cone equations follow by choosing $(i,vu)$ for an arrow $v:j\to j^{\prime }$. This construction is inverse to restriction and acts identically on vertex maps, proving the cone restriction along an initial functor isomorphism.

Terminal objects in the two cone categories therefore correspond. Whenever the $\mathcal{I}$-shaped limit exists,naturally in $D$. Equivalently, the triangle formed by precompositionand the two limit functors commutes up to natural isomorphism.

For the converse, suppose this commutation holds for $\mathcal{C}={\mathbf{Set}}^{\mathrm{op}}$. Passing to opposite categories says that restriction along $F^{\mathrm{op}}$ preserves all set-valued colimits. Fix $j\in \mathcal{J}$ and take the representable functorIts colimit is a singleton: the category of its elements has the initial object $(j,1_j)$. The restricted colimit iswhose elements are precisely the connected components of $(F\downarrow j)$. By the assumed comparison this set is also a singleton. Thus $(F\downarrow j)$ is nonempty and connected for every $j$, so $F$ is initial.

Let $q:C_n\to \mathrm{coker}{d}_{n+1}$. Since $d_n{d}_{n+1}=0$, there is an induced ${\bar{d}}_n:\mathrm{coker}{d}_{n+1}\to C_{n-1}$. In an abelian category,The left side is the usual homology construction and the right side is its construction in the opposite category, so the definition of homology is self-dual.