  We consider the many-to-many bipartite matching problem in the presence of
two-sided preferences and two-sided lower quotas. The input to our problem is a
bipartite graph G=(A U B, E), where each vertex in A U B specifies a strict
preference ordering over its neighbors. Each vertex has an upper quota and a
lower quota denoting the maximum and minimum number of vertices that can be
assigned to it from its neighborhood. In the many-to-many setting with
two-sided lower quotas, informally, a critical matching is a matching which
fulfils vertex lower quotas to the maximum possible extent. This is a natural
generalization of the definition of critical matching in the one-to-one setting
[Kavitha T., FSTTCS 2021]. Our goal in the given problem is to find a popular
matching in the set of critical matchings. A matching is popular in a given set
of matchings if it remains undefeated in a head-to-head election with any
matching in that set. Here, vertices cast votes between pairs of matchings. We
show that there always exists a matching that is popular in the set of critical
matchings. We present an efficient algorithm to compute such a matching of the
largest size. We prove the popularity of our matching using a dual certificate.
