Would someone let me know the answer and the proof or counter example of the following question?

Suppose the category C has a pullback for every pair of morphism (f : X -> Y, g : W -> Y). Let K be the full subcategory of the functor category Func(C,Set) whose objects are pullback perserving functors. Is K ccc? (If so, how I can show this?)

The answer is no. First note that K is closed under products in the functor category. Also, it contains all the representable functors; so, if it were cartesian closed, the exponential G^F would have to be given by G^F(c) \cong nat((c,-),G^F) \cong nat((c,-)\times F,G) i.e. K would have to be closed under exponentials in [C,Set]. However, it isn't in general. For a simple counterexample, let C be the category with five objects a,b,c,d,e and six non-identity morphisms a --> b, a --> c, b --> d, c --> d, a --> d, a --> e ; note that C has just one nontrivial pullback square a -----> b | | | | v v c -----> d Let F be the functor given by F(a) = F(b) = F(c) = F(d) = \emptyset, F(e) = {*}, and let G be F + F. Then (taking the above definition of G^F) G^F(a) has two elements, but G^F(b), G^F(c) and G^F(d) are singletons. Peter Johnstone