In mathematics, specifically category theory, the interchange law (or exchange law) regards the relationship between vertical and horizontal compositions of natural transformations.
Let formula_1 and formula_2 where formula_3 are functors and formula_4 are categories. Also, let formula_5 and formula_6 while formula_7 and formula_8 where formula_9 are natural transformations. For simplicity's and this article's sake, let formula_10 and formula_11 be the "secondary" natural transformations and formula_12 and formula_13 the "primary" natural transformations. Given the previously mentioned, we have the interchange law, which says that the horizontal composition (formula_14) of the primary vertical composition (formula_15) and the secondary vertical composition (formula_15) is equal to the vertical composition (formula_15) of each secondary-after-primary horizontal composition (formula_14); in short, formula_19. It also appears in monoidal categories wherein classical composition (formula_14) and the tensor product (formula_21) take their places in lieu of the horizontal composition and vertical composition partnership and is denoted formula_22. 
The word "interchange" stems from the observation that the compositions and natural transformations on one side are switched or "interchanged" in comparison to the other side. The entire relationship can be shown within the following diagram.
If we apply this context to functor categories, and observe natural transformations formula_5 and formula_6 within a category formula_25 and formula_7 and formula_8 within a category formula_28, we can imagine a functor formula_29, such that
the natural transformations are mapped like such:
functors are also mapped accordingly: