A comment on Zhen Lin's reply to Andrej:
To begin, consider a category C with finite limits. Suppose C has an internal category U such that the externalisation of U as a C-indexed category (or category fibred over C) is equivalent to the self-indexing of C. Since U is locally small as a C-indexed category, the self-indexing of C has the same property, so we deduce that C is locally cartesian closed.
We have a universal fibration el U -> ob U (by restricting the fibration mor U -> ob U x ob U), so it follows that every object X admits a monomorphism X -> el U. Now, if we add the assumption that C (or U) is well-powered as a C-indexed category ...
Just to remark that there is no need to add any assumption here. If a locally cartesian closed category has a "universal fibration" as you call it (called "generic family" in the reference below), then it is degenerate. This appears in: Pitts, Andrew M.; Taylor, Paul. A note on Russell's paradox in locally Cartesian closed categories. Studia Logica 48 (1989), no. 3, 377?387. (MR1059248) With best wishes, Alex -- Alex Simpson, LFCS, School of Informatics, Univ. of Edinburgh, UK Email: Alex.Simpson@ed.ac.uk Tel: +44 (0)131 650 5113 Web: http://homepages.inf.ed.ac.uk/als Fax: +44 (0)131 651 1426 -- The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]