Here a relation$r\colon R\to X\times Y$ is left (respectively right) punctual if $(id_X,0)\colon X\to X\times Y$ (respectively $(0,id_Y)\colon Y\to X\times Y$) factors through $r$.

Example

A variety of algebras is a subtractive category if and only it is a subtractive variety in the sense of Ursini, meaning its theory contains a binary operation $s$ and constant $0$ such that $s(x,x)=0$ and $s(x,0)=x$.

Here $Pt(B)$ is the pointed category whose objects are triples $(A\in C,\alpha\colon A\to B,\beta\colon B\to A)$ such that $\alpha\circ\beta=id_B$. Morphisms $(A,\alpha,\beta)\to(A',\alpha',\beta')$ are morphisms $f\colon A\to A'$ such that $\alpha'\circ f=\alpha$ and $f\circ\beta=\beta'$.