The concept of regular category [1] has several 2-dimensional analogues depending upon which special arrows are chosen to mimic monics. Here, the choice of the conservative arrows, leads to our notion of faithfully conservative bicategory K in which two-sided discrete fibrations become the arrows of a bicategory F = DFib( K) . While the homcategories F(B,A) have finite limits, it is important to have conditions under which these finite “local” limits are preserved by composition (on either side) with arrows of F . In other words, when are all fibrations in F flat? Novel axioms on F are provided for this, and we call a bicategory F modulated when F op is such a F . Thus, we have constructed a proarrow equipment ( ) ∗: → M (in the sense of [28]) with M = F coop . Moreover, M is locally finitely cocomplete and certain collages exist [23]. In the converse direction, if M is any locally countably cocomplete bicategory which admits finite collages [23], then the bicategory M ∗ of maps in M is modulated. (Recall from [26, p 266], that a 1-cell in a bicategory is called a map when it has a right adjoint.)