Remarks on the Tripos To Topos Construction: extensionality, comprehensions, quotients and cauchy-complete objects

We give a description of the Tripos To Topos construction in terms of four free constructions. We prove that these compose up to give a free construction from the category of triposes and logical morphisms to the category of toposes and logical functors. Then we show that other similar constructions, i.e. the one given by Frey in \cite{frey} and that of Carboni in \cite{carbons} are instances of this one.