In Lee's book Smooth manifolds he introduces the following proposition
Proposition 3.14 (The Tangent Space to a Product Manifold). Let $M_1,\dots, M_k$ be smooth manifolds, and for each $j$, let $\pi_j : M_1 \times \dots \times M_k \to M_j$ be the projection onto the $M_j$ factor. For any point $p = (p_1, \dots, p_k) \in M_1 \times \dots \times M_k$ the map $$\alpha : T_p(M_1 \times \dots \times M_k) \to T_{p_1}M_1 \oplus \dots \oplus T_{p_k}M_k$$ defined by $$\alpha(v)=(d(\pi_1)_p(v), \dots,d(\pi_k)_p(v))$$ is an isomorphism.
He leaves the proof as an exercise which I couldn't do, but found the solution to. On the next page he states that
Once again, because the isomorphism is canonically defined, independently of any choice of coordinates, we can consider it as a canonical identification, and we will always do so. Thus, for example, we identify $T_{(p,q)}(M \times N)$ with $T_pM\oplus T_qN$ and treat $T_pM$ and $T_qN$ as subspaces of $T_{(p,q)}(M \times N)$.
This is where I'm getting confused. I don't understand what is he trying to communicate here?
In the solution for the problem for the case where we only have two manifolds $M$ and $N$ it's first shown that $\alpha$ is linear and then that it has an inverse $\beta$ as $$\beta : T_p M \oplus T_q N \to T_{(p,q)} (M \times N)$$ defined by $$\beta(u,w)=d(\iota_M)_p(v) +d(\iota_N)_q(w)$$ where the inclusions are as $\iota_M : M \to M\times N$ and map $M$ to $M \times \{q\}$ and likewise for $\iota_N$.
Now I think there is a connection here the way that the inclusions are defined and what Lee stated about the canonical isomorphism, but I don't see how? I've tried to find some simpler examples of this online, but there isn't any.
If anyone could explain the motivation for defining the inverse $\beta$ with the inclusions as it is that would be very much appreciated.