Withe

is a primitive operator which forms a hook. Called dyadically with function operands  and , it uses   monadically to pre-processes the right argument before applying   to the given left argument and pre-processed right argument. Called monadically with operands  and , it uses   monadically to pre-processes the argument before applying   to the argument and pre-processed argument.

In usage,  is equivalent to , and   is equivalent to. Thus, Withe can be defined as the dop  and is equivalent to the derived function.

Withe first appeared in A Dictionary of APL, but substituted by the 2-train in J after the proposal in Iverson and McDonnell's paper Phrasal Forms. It is present in Kap as Compose and in BQN as After.