Under

, or Dual, is a primitive dyadic operator which takes two function operands. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.

Model
In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):

Examples
The following example solves a LeetCode problem which asks to add two numbers given as reversed digit lists: First, each argument is reversed, then decoded from base 10. The addition is performed. Finally, the result is encoded to base 10 and reversed.

Close composition
In SHARP APL and J, Under is implemented as a close composition, meaning that (using SHARP syntax)  has the overall function rank of. This allows  to serve the purpose of Each from a nested APL. J uses  for the close form and   for the rankless form.

History
The most widely known form of Under for much of APL's history was SHARP APL's Dual operator, written  and defined as a close composition. It was presented by Ken Iverson and Bob Bernecky with the name "with" in 1980, and implemented for only a small number of right operands in 1981. The name "Dual" was used in Rationalized APL in 1983, and the full SHARP implementation using function rank was completed just after. Iverson changed the name to Under for his 1987 A Dictionary of APL, and J uses this name as well. It includes both a close form  like SHARP and a non-close form.

However, Iverson had defined the Dual operator with glyph  in his 1978 paper Operators and Functions, using a non-close form because function rank had not yet been invented. NARS featured this operator in 1981, preceding SHARP's limited implementation. Roger Hui proposed Under with the same definition as a potential addition to Dyalog APL at Dyalog '15, and it was added to both Extended Dyalog APL and dzaima/APL in 2018.

Structural Under was developed by Marshall Lochbaum around 2017, and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later dzaima would implement  and   in the same spirit, now featured in dzaima/APL and dzaima/BQN. The language BQN has included structural and computational Under, using the glyph, since its initial design in 2020. Structural Under also appears in Kap and Uiua, and is implemented for a small number of cases in J version 9.5 (2023).