Universal updates for symmetric lenses

Asymmetric c-lenses are the special cases of asymmetric d-lenses (also called delta lenses) whose updates satisfy a universal property which in many applications ensures "least-change". There has therefore been hope that symmetric c-lenses might characterize those symmetric dlenses which satisfy a similar universal property. This paper begins an analysis of symmetric c-lenses and their relationship to symmetric dlenses and explains why the authors do not expect symmetric c-lenses, that is, equivalence classes of spans of c-lenses, to be central to developing universal properties for symmetric lenses. Instead, we consider cospans of c-lenses and show that they generate symmetric c-lenses with an appropriate universal property. That property is further analysed and used to motivate proposed generalisations to obtain universal, least-change, properties for symmetric d-lenses. In addition we explore how to characterise the symmetric d-lenses that arise from cospans of c-lenses among all symmetric d-lenses.