Dear Categorists, I have three loosely related requests for references for constructions that I am using in some of my current work. First, does someone know of a publication, in which it is proved that for a category I the 2-functor [I, -] : Cat -> Cat given by post-composition extends to a map on the (large) fibration Fib -> Cat? Alternatively, I would also be happy to have references for the fact that [I, p] : [I, E] -> [I, B] is a fibration for a fibration p : E -> B and a proof that [I, -] preserves strict 2-pullbacks, as these can be put together to obtain the above result. Second, I need to construct exponents in general presheaves I^op -> C. In particular, I am interested in the case where I is the poset ?? of finite ordinals. I am able to show that [??^op, C] is Cartesian closed, if C has finite limits and is Cartesian closed. In this case, the exponent s => t for s, t : ??^op -> C is given as follows. First, for every natural number n, one defines a functor Sn : N x N^op -> C, where N is the poset of ordinals less or equal to n, by putting Sn(m, k) = t(k)^{s(m)} and Sn(m' <= m, k <= k') = t(k <= k')^{s(m' <= m)}, in which (-)^(-) : C x C^op -> C is the exponent in C. We can then define the exponent s => t as the end (s => d)(n) = \int_{m in N} Sn(m, m), which exists in C because N is a finite category and C has finite limits. It takes a bit of effort to check this, but the construction gives us indeed exponents in [??^op, C]. Did someone see this construction before or is aware of another way to construct exponents in presheaf categories like [??^op, C]? Finally, the previous construction can be also applied to the fibres of the fibration [??^op, p] : [??^op, E] -> [??^op, B], if p : E -> B is a fibred CCC with fibred finite limits. Thus, one also obtains that the fibration [??^op, p] is a fibred CCC. Again, the question is, whether someone has seen something like this before. Thank you all very much in advance. Best, Henning [For admin and other information see: http://www.mta.ca/~cat-dist/ ]