We consider pre-exponentiable objects of a pre-cartesian double category D, i.e., objects Y such that the lax functor - x Y: D --> D has a right adjoint in the 2-category LxDbl of double categories and lax functors. When D has 2-glueing, we show that Y is pre-exponentiable in D if and only if Y is exponentiable in D_0 and - x Y is an oplax functor. Thus, such a D is pre-cartesian closed as a double category if and only if D_0 is a cartesian closed category. Applications include the double categories cat, pos, spaces, loc, and topos, whose objects are small categories, posets, topological space, locales, and toposes, respectively.
Keywords: double category, exponentiability, cartesian closed, glueing
2020 MSC: 18N10,18D15, 18F10, 18F70, 54C35, 16B35
Theory and Applications of Categories, Vol. 35, 2020, No. 32, pp 1208-1226.