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.
Excellent observation! Thus C is a model for Martin-Loef type theory with Type : Type (as interpreted by U). It is known since Girard's paradox from 1970 that this type theory is inconsistent, i.e. that every type is inhabited. Thus, in particular, all identity types are inhabited which renders C trivial. For a more categorical account of the inconcistency of type theory with Type:Type see A note on Russell's paradox in locally cartesian closed categories Andrew M. Pitts & Paul Taylor Studia Logica 48 (3):377 - 387 (1989) which is much simpler than Girard's original proof. But I don't understand some of your subsequent arguments.
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.
why? and if so what is the point? in a topos for every object X there is a mono 0 -> X
Now, if we add the assumption that C (or U) is well-powered as a C-indexed category, then C must be an elementary topos. But then the existence of el U implies that the internal logic of C is inconsistent, so C must be the degenerate topos.
You can't derive wellpoweredness because C lacks (nice) quotients, isn't it? Moreover, for showing that a topos with a universal family is trivial you need something like a proof that Type:Type is inconsistent. But I think that quite correctly you spotted the inconsistency `a la Russell, namely that there is not a set of all sets (or a type of all types). Thomas Streicher [For admin and other information see: http://www.mta.ca/~cat-dist/ ]