### Lex total categories and Grothendieck toposes

In the very valuable volume Sketches of an Elephant, A Topos Theory Compendium, Volume 1, Peter Johnstone mentions a lecture of Andre Joyal at the 1981 Cambridge Summer Meeting where he listed seven different descriptions of 'what a topos is like'. His fifth one was

(v) 'A topos is a totally cocomplete object in the meta-2-category of finitely complete categories'.