Preimage resistance hash in digital signature
I'm having a hard time parsing your question.

If C can compute pre-images on $H$, then he can forge digital signatures from Bob. All C needs is one signature from Bob.

Bob signs message $m$, the signature is $S={H(m)}^d\bmod n$. All C needs to do is find an $m'$ where $H(m)=H(m')$. The forged message and signature will be $m'$ and the original $S$.