In Lee's Introduction to Smooth Manifolds 2nd Edition, item (b) of Proposition 2.15 reads as follows:
Every finite product of diffeomorphisms between smooth manifolds is a diffeomorphism.
What exactly does he mean by "finite product of diffeomorphisms"?
If $f_i:M_i\to N_i$ denotes a map of smooth manifolds for $i=1,\ldots, k$, then as stated above, the product map $f_1\times\cdots\times f_k:\prod_i M_i\to \prod_i N_i$ is defined by $(f_1\times\cdots\times f_k)(x_1,\ldots, x_k)=(f(x_1),\ldots, f(x_k))$.
Indeed, the product of finitely many manifolds is again a manifold, so everything here makes sense.