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 $\mathbf{Rel}$.

翻译：最初在项图重写的代数方法背景下引入的gs-幺半范畴概念，在过去几十年中以不同名称数次出现。它们可被视作箭头为广义关系的对称幺半范畴，具有足够结构以讨论定义域与偏函数，但结构弱于笛卡尔双范畴。本文旨在实现三个目标。首先，通过为箭头添加预序结构来扩展gs-幺半性的原始定义，进而提出所谓的预序笛卡尔范畴。其次，证明（预序丰富）gs-幺半范畴自然地同时以克莱斯利范畴和跨范畴的形式出现，并探讨了这两种形式化之间的关系。最后，我们分别呈现关于米田嵌入与函子完备性的两个定理，后者导出了从预序笛卡尔范畴到$\mathbf{Rel}$的松弛函子的完备性结果。