On page 30 of Proofs and Types by Girard, Taylor, and Lafont, it is claimed that that the identity axiom for sequent calculus:C  Chas a converse relation with the cut rule:$$\frac{\vec{A} \vdash C,\vec{B}  \qquad \vec{A'},C \vdash \vec{B'}}       {\vec{A},\vec{A'} \vdash \vec{B},\vec{B'}}$$The explanation it gives is that the in the identity axiom, C on the left is stronger than C on the right. With the cut rule, it claims, C on the right is stronger than C on the left. While the claim about the identity axiom seems to follow from basic notions of sequents, I do not understand the claim about the cut rule. Could someone explain this?
converse relationship between the cut rule and the identity axiom
lo.logic;proof theory;sequent calculus