Algebraic exponentiation for categories of Lie algebras

For a category C, D. Bourn’s categories of points (categories of split epimorphisms with fixed codomain) can be defined as categories of the form ((B,1B)↓(C↓B)) for some B in C. A categorical-algebraic concept of exponentiation, namely, right adjoints for the pullback functors between D. Bourn’s categories of points, was introduced and studied in the author’s Ph.D. Thesis. We show for every category of Lie algebras over a fixed commutative unital ring, that all exponents exist.