On functor double -categories

In this paper, we study double -categories of double functors. To this end, we exhibit the cartesian closed structure of the -category of double -categories and various localizations. We prove a theorem that characterizes the companions and conjoints in functor double -categories via the notion of companionable and conjointable 2-cells in double -categories. Moreover, we show that under suitable conditions, functor double -categories are horizontally closed. Throughout the paper, we highlight a few applications to (, 2) -category theory and indexed exponentiability.