Given a 3SAT problem. The question being: 'This Problem has exactly K Solutions'?
Now, lets say K=1 (without loss of generality).
If the problem has a exactly 1 solution and the answer is True. So, the partial certificate is that 1 solution. the certificate is partial as it shows there is a solution but without guarantee of exactly 1.
If the answer is False. Then either it has 0 or more than 1 solution.
(a) In case of more than 1 solution, the certificate is any of the 2 or more solutions. This is a complete certificate.
(b) In case of 0 solution, the problem is same as asking does the given SAT have a solution. The certificate being same in 1.
Thus, the problem seems to require a certificate from both NP and co-NP versions. It seems to mean problem lies in NP and co-NP. But then it also is NP-hard. So doesn't that imply NP=co-NP?! What am I missing.