Dear Steve On 1 Nov 2025, at 4:32 am, Steven Vickers <s.j.vickers.1@bham.ac.uk> wrote: If you forget D and t, T looks very like a limit sketch. C is presented by the graph (sorts and operators) and commutative diagrams (equations), while the objects b of B are for the cones. v applied to the fibre of u over b gives the diagram, w(b) the vertex, and τ provides the cone itself. Hence all that is still needed is to require that the cones are limit cones. As far as I can see, that is what the diagram at the top of p.2 is saying in the case t=1. Yes, that is correct. (I should have been less opaque!) A limit sketch yields a T with no D or t (i.e. t = 1_B), and with B discrete. I expect we obtain the same models by restricting u to be a fibration. However, I'm still puzzled by the role of t. Can you give any intuitions about just what kind of theory the general T presents? Let's say, in the finite case and for X with equalizers and finite products. Is it any more general than lex theories? I think I was trying to include the case where cones were replaced by the cylinders of Freyd-Kelly [Categories of continuous functors I, JPAA (1972)] and the case of "rules" of John Isbell [General functorial semantics. I. Amer. J. Math. 94 (1972), 535--596]. However, I cannot remember to what extent that works. I think, therefore, there should be enough there to give me the objects of models that I'm looking for. Is there a publication I can reference? Joseph Helfer's suggestions are relevant: "in my [paper] Definition 2.14, I cite Street's Cosmoi of internal categories, §9.14, where they are discussed." Perhaps also Jean Bénabou's [Théories relatives à un corpus. C. R. Acad. Sc. Paris, t. 281 (17 novembre 1975)]. Section 5 of my [Conspectus of variable categories, JPAA 21 (1981) 307--338] could be of some value because categories A can be brought into a bicategory K by tensoring (copowering) A with the terminal object of K. Ross