Dear Steve "Studied" is probably the wrong word for http://science.mq.edu.au/~street/Sketch.pdf which is F. Complete objects relative to a theory (June 1976) on my Publication page. Others will have better suggestions. Ross On 30 Oct 2025, at 3:05 am, Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote: An nLab page https://ncatlab.org/nlab/show/cartesian+object<https://ncatlab.org/nlab/show/cartesian+object> defines an object of X of a 2-category K to be “cartesian” iff the 1-cell X -> 1 and the diagonal X -> X^2 both have right adjoints. This assumes K has finite 2-products. If, further, K has finite PIE-limits, hence finite powers (cotensors) X^C, then it seems reasonable to define X to be “lex” if every diagonal X -> X^C has a right adjoint. Unless I’ve made a mistake, in the 2-category of categories this characterises the lex categories. Have these lex objects in a 2-category been studied? Steve Vickers.