Connection with exactness in $\Ab$

[002A] Remark 1·r (Connection with exactness in $\Ab$).

Conditions ), ), and ) apply in every representable 2-category, and they are in general not equivalent to each other. A fortiori this is true in every 1-category.
Considering a 1-category $K$ as a 2-category $\mathbb K$ without nontrivial 2-morphisms, conditions ), ) make sense; so, (cf. Hilton (Hilton, 1966)) in $\Ab$ a square is exact in the sense ) if and only if the sequence $A\to X\to B$ is exact at the object $X$.
Since $\Ab$ embeds in $\Cat$ [an abelian group is a one-object category, TN], the square above can be regarded in $\Cat$; and it is exact in the 2-category $\Cat$ if and only if $A\to X\to B\to 0$ is exact in $X$ and in $B$.
The calculus of exactness in $\Cat$ is thus an enlargement of the one in $\Ab$; in this spirit, the absolute exact squares (e.g. adjunctions and Yoneda squares) generalise the split exact sequences, and $\tilde\vf$ represents the homology of the square $\vf$.