It is possible to perform diagram chases even in categories in which the objects are not sets, with a mathematical sleight-of-hand using something called members.

Call an arrow $x$ with codomain $a\beta \x88\x88A$ a member of $a$, written $x{\beta \x88\x88}_{m}a$, and define $x\beta \x89\u2018y$ for two members of $a$ to mean there are epimorphisms $u,v$ with $xy=yv$. One can check this is an equivalence relation on the set of members of $a$. We can then think of members of $a$ as equivalence classes of arrows to $a$, with this relation.

Each object $a$ has a zero member (the equivalence class of the zero arrow $0\beta \x86\x92a$). Each member $x{\beta \x88\x88}_{m}a$ also has a "negative", denoted $\beta \x88\x92x$.

Rules for chasing diagrams

For the members in any abelian category:

$f:a\beta \x86\x92b$ is a monomorphism if and only if for all $x{\beta \x88\x88}_{m}a$, $fx\beta \x89\u20180$ implies $x\beta \x89\u20180$;

$f:a\beta \x86\x92b$ is a monomorphism if and only if, for all $x,{x}^{\beta \x80\xb2}{\beta \x88\x88}_{m}a$, $fx\beta \x89\u2018f{x}^{\beta \x80\xb2}$ implies $x\beta \x89\u2018{x}^{\beta \x80\xb2}$;

$g:b\beta \x86\x92c$ is an epimorphism if and only if for each $z{\beta \x88\x88}_{m}c$ there is $y{\beta \x88\x88}_{m}b$ with $gy\beta \x89\u2018z$;

$h:r\beta \x86\x92s$ is the zero arrow if and only if, for all $x{\beta \x88\x88}_{m}r$, $hx\beta \x89\u20180$;

A sequence $a\stackrel{f\phantom{\rule[-.25em]{0ex}{0ex}}}{{\textstyle \beta \x86\x92}}b\stackrel{g\phantom{\rule[-.25em]{0ex}{0ex}}}{{\textstyle \beta \x86\x92}}c$ is exact at $b$ if and only if $gf=0$ and to every $y{\beta \x88\x88}_{m}b$ with $gy\beta \x89\u20180$ there exists $x{\beta \x88\x88}_{m}a$ with $fx\beta \x89\u2018y$;

(Subtraction) Given $g:b\beta \x86\x92c$ and $x,y{\beta \x88\x88}_{m}b$ with $gx\beta \x89\u2018gy$, there is a member $z{\beta \x88\x88}_{m}b$ with $gz\beta \x89\u20180$; moreover, any $f:b\beta \x86\x92d$ with $fx\beta \x89\u20180$ has $fy\beta \x89\u2018fz$ and any $h:b\beta \x86\x92a$ with $hy\beta \x89\u20180$ has $hx\beta \x89\u2018\beta \x88\x92hz$.