What is a tangent bundle? (Aubin) Here's what I read in A Course in Differential Geometry by Thierry Aubin.

2.5. Definition. The tangent bundle $T(M)$ is $\bigcup_{P\in M} T_P(M).$

And then

2.6. Definition. Let $\Phi$ be a differentiable map of $M_n$ into $W_p$ (two differentiable manifolds). Let $P\in M_n,$ and set $Q=\Phi(P).$ The map $\Phi$ induces a linear map $(\Phi_*)_P$ of the tangent bundle $T_P(M)$ into $T_Q(W)$ defined by $$[(\Phi_*)_PX](f)=X(f\circ\Phi);$$
here $X\in T_P(M),\;(\Phi_*)_PX\in T_Q(W)$ and $f$ is a differentiable function in a neighbourhood $\theta$ of $Q.$ We call $(\Phi_*)_P$ the linear tangent mapping of $\Phi$ at $P.$

I don't understand why the author calls $T_P(M)$ a tangent bundle in the second definition. Is it a mistake? From the first definition, a tangent bundle is the union of all tangent spaces over all points of the manifold. And $T_P(M)$ is just one tangent space, at a particular point $P$.
And an additional question: Should I be worried whether the union in the first definition is disjoint or not? After a moment's thought, I believe it might turn out not to be according to the previous definitions.

Definition 2.6 has a typo; $(\Phi_\ast)_P$ is a map of tangent spaces, not tangent bundles (although all of the $(\Phi_\ast)_P$ combine to form a bundle map between tangent bundles).
The tangent bundle is the disjoint union of the tangent spaces:
$$TM = \coprod_{P \in M} T_P M.$$
It has the topology of a smooth manifold in the following manner. Let $(U_\alpha, \phi_\alpha)$ be an atlas for $M$, and let $\pi: TM \longrightarrow M$ be the natural projection, i.e. if $(P, v) \in T_P M \subset TM$, then $\pi(P, v) = P$. Then we get an atlas $(\pi^{-1}(U_\alpha), \tilde{\phi}_\alpha)$ for $TM$, where
$$\tilde{\phi}_\alpha(P, v) = (\phi_\alpha(P), v).$$
