  Originally introduced in the context of the algebraic approach to term graph
rewriting, the notion of gs-monoidal category has surfaced a few times under
different monikers in the last decades. They can be thought of as symmetric
monoidal categories whose arrows are generalised relations, with enough
structure to talk about domains and partial functions, but less structure than
cartesian bicategories. The aim of this paper is threefold. The first goal is
to extend the original definition of gs-monoidality by enriching it with a
preorder on arrows, giving rise to what we call oplax cartesian categories.
Second, we show that (preorder-enriched) gs-monoidal categories naturally arise
both as Kleisli categories and as span categories, and the relation between the
resulting formalisms is explored. Finally, we present two theorems concerning
Yoneda embeddings on the one hand and functorial completeness on the other, the
latter inducing a completeness result also for lax functors from oplax
cartesian categories to $\mathbf{Rel}$.
