What is the Certificate for Set Cover?
The set cover is not a decision problem, so it doesn't necessarily have a certificate.  Rather, it is an optimization problem.

To make it a problem in NP (a decision problem, with a certificate), you need to apply the standard trick to convert an optimization problem to a decision problem: introduce a budget.  Thus, the decision budget is: given a collection of sets and an integer $k$, determine whether there exists a subset of $\le k$ sets whose union is all of $\{1,\dots,m\}$.  That is a decision procedure, and does have a certificate: the subset of size $k$ is the certificate.

See also Optimization version of decision problems and standard textbooks.