I have a result that holds for cocomplete and finitely complete categories such that pullbacks preserve directed colimits, by which I mean $A \times\_B (\operatorname{colim}\_{i \in I} C\_i) = colim\_{i \in I} (A \times\_B C\_i)$, where $I$ is a directed category. 


Thanks to a result by Adamek and Rosicky [locally presentable and accessible categories, Proposition 1.59] , I know that this holds in any locally (finitely) presentable category. 


Is there any other category that satisfies those conditions?



Cocomplete and finitely complete category with nice pullbacks that is not locally presentable
If you look at the article on [quasitoposes](https://ncatlab.org/nlab/show/quasitopos#examples), which are locally cartesian closed and therefore satisfy your exactness condition, you'll find a number of examples that are not locally presentable. For example, the category of pseudotopological spaces, the category of bornological sets, the category of equilogical spaces, and (I think) the category of quasitopological spaces in the sense of Spanier. I might think of more (preferably a whole class of more) later. 


