External direct product of formal group laws

Definition
Suppose $$R$$ is a commutative unital ring. Suppose $$F_1$$ and $$F_2$$ are defining ingredient::formal group laws over $$R$$ of dimensions $$d_1$$ and $$d_2$$ respectively. The external direct product of $$F_1$$ and $$F_2$$, which we denote $$F_1 \times F_2$$, is a formal group law of dimension $$d_1 + d_2$$, given by:

$$(F_1 \times F_2)(x_1,x_2,\dots,x_{d_1},x_{d_1 + 1},\dots,x_{d_1+d_2},y_1,y_2,\dots,y_{d_1},y_{d_1+1},\dots,y_{d_1+d_2})$$

$$ = (F_1(x_1,x_2,\dots,x_{d_1},y_1,y_2,\dots,y_{d_1}),F_2(x_{d_1 + 1},\dots,x_{d_1+d_2},y_{d_1+1},\dots,y_{d_1+d_2}) )$$

In other words, the first $$d_1$$ coordinates of $$F_1 \times F_2$$ are obtained by applying $$F_1$$ to the first $$d_1$$ coordinates of the two inputs, and the last $$d_2$$ coordinates of $$F_1 \times F_2$$ are obtained by applying $$F_2$$ to the last $$d_2$$ coordinates of the two inputs.