[0027] Example 1·q.

Applying the previous criteria for exactness Definition 1·b [001M], 1·c·a [001O], 1·c·b [001P], 1·c·c [001Q], 1·d·a [001S], 1·d·b [001T], 1·f [001V] and its opposite, Remark 1·i [001Y],Theorem 1·j [001Z], Theorem 1·m [0023] (all equivalent to each other in $\Cat$) as well as the composition, dualisation, exponentiation criteria, we immediately obtain the following examples:

  • (EE1) the squares are exact (this is evident using Remark 1·i [001Y]. Following Street (Street, 1974) this is equivalent to the Yoneda lemma; we will call these squares Yoneda squares.
  • (EE2) a comma square is exact.
  • (EE3) a co-comma square is exact.
  • (EE4) the square is exact if and only if $F$ is fully faithful.
  • (EE5) the square is exact if and only if $F$ is co-fully faithful.
[0028] Translator’s Note 1·q·a.

A “co-fully faithful” functor is a functor $F$ with the property that each pre-composition $_\circ F$ is fully faithful. (Thanks to Nathanael Arkor for the suggestion!)

  • (EE6) a square is exact if and only if $\epsilon$ is the counit of an adjunction $L\dashv U$;
  • (EE7) a square is exact if and only if $\eta$ is the unit of an adjunction $L\dashv U$;
  • (EE8) more generally, the square is exact if and only if $\epsilon$ is the counit of a relative adjunction $L\overset{Y}\dashv U$ (i.e., $L$ is a partial left adjoint to $U$, with respect to $Y$);
  • (EE9) a square is exact if and only if $\langle V,\vf\rangle$ is an absolute left extension of $S$ along $T$ (cf. §5.1);
  • (EE10) a square is exact if and only if $\langle U,\vf\rangle$ is an absolute right extension of $S$ along $T$.
[0029] Remark 1·q·b.

Condition EE8 can be dualised to relative right adjoints. From EE10 we deduce the formal characterisation of adjoints: the counit of an adjunction exhibits the left adjoint as the absolute right extension of the identity along the right adjoint, and dually for the unit.