nLab
Godement product

The Godement product of two natural transformations between appropriate functors is their horizontal composition? as 2-cells in the 2-category Cat of categories, functors and natural transformations.

For categories $A,B,C$, if $\alpha :F_1\to G_1:A\to B$ and $\beta :F_2\to G_2:B\to C$ are natural transformations of functors, the components $(\alpha *\beta {)}_M$ of the Godement product $\alpha *\beta :F_2\circ F_1\to G_2\circ G_1$ are defined by any of the two equivalent formulas:

$$(\beta *\alpha {)}_M={\beta }_{F_{2}M}\circ G_1({\alpha }_M)$$(\beta * \alpha)_M = \beta_{F_2 M}\circ G_1(\alpha_M)

$$(\beta *\alpha {)}_M=G_2({\alpha }_M)\circ {\beta }_{F_{1}M}$$(\beta * \alpha)_M = G_2(\alpha_M)\circ\beta_{F_1 M}

The Godement product is strictly associative (so that $\mathrm{Cat}$ is a strict 2-category).

The interchange law in (general) 2-categories (which in the case of $\mathrm{Cat}$ boils down to assertion that the two formulas above are equivalent) is also sometimes called Godement interchange law.