@DavidRoberts And the category of diffeological spaces is cartesian closed. I am wondering whether diffeological spaces are a closure of some sort of the category of smooth manifolds ? I can't find the answer in the nLab. It is only explained why the inclusion functor is full and faithful.

@varkor Good point. the category of general topological spaces is a much simpler example. A Hausdorff topological space is exponentiable if and only if it is locally compact.

The category of (small) semicategories also called non-unital categories (ncatlab.org/nlab/show/semicategory) should answer your question (I am not sure to understand the end of your sentence).