When we talk about arrows in terms of functional programming, we really mean Hughes’ Arrows (or Freyd-categories) a generalization of monads. The term arrow rep

When we talk about arrows in terms of functional programming, we really mean Hughes’ Arrows (or Freyd-categories) a generalization of monads. The term arrow represents a level of abstraction over computations, essentially modeling them as objects with inputs and outputs. Hughes' Arrows extend beyond the traditional notion of functors and monads in handling side effects and common computational patterns in a more structured manner. Arrows are useful because they provide a computation model that can implicitly make use of state without ever exposing that state to the programmer. The programmer can use arrowized computations and combine them to create sophisticated systems. A strong arrow is an arrow A having an extra structure known as strength. The strength for A is a natural transformation, generally denoted by t, which consumes a pair of objects (X,Y) and gives back an arrow in the category. In a concrete manner, the strength has this form: t_X,Y: X x AY -> A(X x Y), called tensorial strength. The strength obeys some particular coherence conditions. A deeper explanation involves the monoidal structure of a category and how the 'strength' aids in elaborating the combination of values in the arrow context.