What is the natural isomorphism between the tangent space of a product manifold and the product of the tangent spaces?

Question

Let $M$ and $N$ be smooth manifolds and $p\in M$, $q \in N$. What is then a natural isomorphism for

$$T_{(p,q)}(M\times N) \cong T_pM \times T_qN ?$$

Answer 1

If you see the tangent space in $(p,q)$ as the velocities $\gamma'(0)$ of the curves $\gamma:(-\epsilon, \epsilon) \rightarrow M \times N$ such that $\gamma(0)=(p,q)$ the isomorphisms follows simply from the fact that the projections of $\gamma$ on $M$ and $N$ are two curves $\alpha$ and $\beta$ such that $\gamma'(0)=(\alpha'(0), \beta'(0))$, and that a curve to a product space is uniquely determined by his projections to the single factors.