converse relationship between the cut rule and the identity axiom
Girard is talking about intuitionistic sequents, in which sequences to the right of the turnstile have length at most 1. So in the premiss of the cut rule, the sequence of formulas $\vec{B}$ is empty. We can rewrite the cut rule as follows, writing $B^?$ for a sequence which is either empty or contains a single formula.

$$
\frac{\vec{A} \vdash C \qquad \vec{A&#39;},C \vdash B^?}
     {\vec{A}, \vec{A&#39;} \vdash B^?}
$$

Now, it's easy to see why Girard says $C$ on the right is stronger -- it is the only formula that occurs to the right in the first premiss.