The axioms AB5 and AB5*

First fix the small-family operations used below. In an AB3 abelian category, the join of a small family $(B_i\rightarrowtail A{)}_{i\in I}$ is the image of the induced morphism

${\coprod }_{i\in I}{B}_i⟶A\mathrm{.}$

In an AB3* abelian category, let $q_i:A\to A\mathrm{/}{B}_i$ be the quotient maps (The quotient of an object by a subobject). The meet of the family is the kernel of the induced morphism

$A⟶{\prod }_{i\in I}A\mathrm{/}{B}_i\mathrm{.}$

These constructions have the claimed order properties. Indeed, each component $B_i\to A$ factors through the image of ${\coprod }_i{B}_i\to A$, while any common upper bound receives the coproduct map; image minimality (The image is the least subobject through which a morphism factors) therefore makes that image the least upper bound. Dually, the displayed kernel lies in every $B_i$ because $B_i\rightarrowtail A$ is the kernel of $q_i$ (Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel), and every common lower bound is killed by every $q_i$, hence by the product map, so it factors through the displayed kernel. Thus that kernel is the greatest lower bound. For a two-member family these constructions agree with the binary operations of The subobjects of an object in an abelian category form a lattice. The empty join is $0$ and the empty meet is $A$.

An abelian category satisfies AB5 when it satisfies AB3 and for every small directed family of subobjects $(B_i)$ of an object $A$ and every subobject $C\le A$ one has

$\left({\bigvee }_i{B}_i\right)\wedge C={\bigvee }_i(B_i\wedge C)\mathrm{.}$

It satisfies AB5* when it satisfies AB3* and for every small decreasing family of subobjects $(B_i)$ of an object $A$ and every subobject $C\le A$ one has

$\left({\bigwedge }_i{B}_i\right)\vee C={\bigwedge }_i(B_i\vee C)\mathrm{.}$

The joins and meets in these formulas are the small-family constructions above.