Lax completeness for gs-monoidal categories

Originally introduced in the context of the algebraic approach to term graph rewriting, the notion of gs-monoidal categories has surfaced a few times under different monikers in the last decades. They can be concisely described as symmetric monoidal categories with enough structure to represent relations and partial functions, and as such they have been investigated of lately, also due to their connections with bicartesian categories. The aim of this paper is threefold. The first goal is to rephrase the original definition of gs-monoidality in contemporary terms using the graphical formalisms of string diagrams, at the same time highlighting all the basic properties underlying its structure. Then, we show that gs-monoidal categories naturally arise both in terms of Kleisli categories and of span categories, and the relation between the resulting formalisms is thoroughly explored, resulting in the introduction of the novel concept of weakly affine monad. Finally, after considering preorder enrichments on gs-monoidal categories, we present two theorems concerning functorial completeness on the one hand and Yoneda embeddings on the other hand, the former inducing a completeness result also for functors from gs-monoidal categories to $\mathbf{Rel}$.