Relationship between Beth companion notions

Determine the relationship between the notion of Beth companion introduced for strong Beth companions of quasivarieties in CKM2026 and the notion of Beth companion defined for locally finitely presentable categories in the present framework.

Background

The paper compares its categorical notion of strong Beth companion with a notion introduced independently for classes of algebras, especially quasivarieties, in CKM2026. The two approaches use different frameworks: the cited work relies on implicit operations and universal algebra, whereas the present paper develops Beth companions through mono-reflective subcategories, saturated objects, and categorical orthogonality.

The authors explicitly state that the relationship between the two notions has not been resolved in general. They do note that for quasivarieties with the amalgamation property, the notion from CKM2026 is a strong Beth companion in the sense used in the paper.