  The purpose of this article is to compare several versions of bivariant
algebraic cobordism constructed previously by the author and others. In
particular, we show that a simple construction based on the universal
precobordism theory of Annala--Yokura agrees with the more complicated theory
of bivariant derived algebraic cobordism constructed earlier by the author, and
that both of these theories admit a Grothendieck transformation to operational
cobordism constructed by Luis Gonz\'alez--Karu over fields of characteristic 0.
The proofs are partly based on convenient universal characterizations of
several cobordism theories, which should be of independent interest. Using
similar techniques, we also strengthen a result of Vezzosi on operational
derived $K$-theory. In the appendix, we give a detailed construction of virtual
pullbacks in algebraic bordism, filling the gaps in the construction of
Lowrey--Sch\"urg.
