I'm reading Spivak's Mechanics book and he says the following when talking about Hamiltonian Mechanics
Given a Lagrangian $L : TM\to \mathbb{R}$, at each point $a\in M$ the restriction $L_a = L|M_a$, is a smooth function taking the vector space $M_a$ to $\mathbb{R}$. We can therefore consider the derivative
$$D(L_a):M_a\to M_a^\ast$$
By putting together all the maps $D(L_a)$, we obtain a map from $TM$ to $T^\ast M$ called the "fibre derivative" of $L$, though the "fibre-wise derivative" of $L$ might be a better name. For this map we will adopt the notation
$$\mathbf{F}DL:TM\to T^\ast M = \bigcup_{a\in M}D(L_a): M_a\to M_a^\ast.$$
Now, I do understand how does one get each $D(L_a)$ maps, but what means this "putting together all the maps $D(L_a)$"? I'm not understanding how this map $\mathbf{F}DL$ is defined.
My opinion is that it is simply defined by requiring that $\mathbf{F}DL|M_a = D(L_a)$. In other words, if $v\in TM$ is $v=(a,v_a)$, that is, $v\in M_a$, then we have
$$\mathbf{F}DL(a,v) = D(L_a)(v_a)$$
Or equivalently, if $\pi : TM\to M$ is the usual projection we have
$$\mathbf{F}DL(v)=D(L_{\pi(v)})(v).$$
Is this how this fibre derivative is defined? Is this the meaning of putting together all the maps?