Does these MITM-middleware softwares usually verify the cert presented by the server? If not, I guess this could be used to double-MITM the user? If someone ma

Does these MITM-middleware softwares usually verify the cert presented by the server? If not, I guess this could be used to double-MITM the user? If someone manages to redirect traffic by e.g DNS spoof to some server which presents a self-signed certificate for e.g Facebook.com, the government-MITM would just sign that as being Facebook.com.