Consider the convenient category of topological spaces - the space of Compactly Generated Weak Hausdorff spaces. This category is a cartesian closed category.
Now for every obect in this category, introduce the (contravariant) functor $\mathcal{B}$ which sends a space $X$ to the measure space consisting of the space itself along with it's Borel $\sigma$-algebra.
This functor is faithful since continuous maps are measurable in the Borel sigma algebra.
Is it true that the full subcategory of CGWH sitting inside the category of measurable spaces (with measurable maps) is also cartesian closed?