Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$
maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point
$y \in \mathbb{P^1} $ $\textit{without}$ any equivalence
relation imposed. Hence, there is a well defined projection map
$$ \pi_1:H_{0,1}(\mathbb{P}^2, d) \longrightarrow \mathbb{P}^1 \qquad
\pi_1(u,y):= y. $$
Let us define the line bundle
$J := \pi_1^* T\mathbb{P}^1 \longrightarrow H_{0,1}(\mathbb{P}^2, d)$.
In words, the fiber at each point $(u,y)$ is the tangent space of
$\mathbb{P}^1$ at $y$. Now let us define
$$M_{0,1}(\mathbb{P}^2, d) := H_{0,1}(\mathbb{P}^2, d)/PSL(2, C) $$
Where $PSL(2, C)$ is the automorphism group of $\mathbb{P}^1$. The
action of this group lifts to an action on the bundle $J$ and hence
there is an induced bundle $L\rightarrow M_{0,1}(\mathbb{P}^2, d)$.
Also note the following fact: the section of the bundle
$J^* \otimes \mathrm{ev}^*T\mathbb{P}^2\longrightarrow H_{0,1}(\mathbb{P}^2, d)$ given by
$$ (u,y) \longrightarrow du|_y $$
descends to a section of $L^* \otimes \mathrm{ev}^*T\mathbb{P}^2 \rightarrow M_{0,1}(\mathbb{P}^2, d)$.
Here $\mathrm{ev}$ is the evaluation at the marked point $y$.
In words, "taking deriviatve of the curve at the marked point"
is a well defined notion after passing to the quotient.
$\textbf{Question:}$ How does one extend the bundle $L$ to the compactification $\overline{M}_{0,1}(\mathbb{P}^2, d)$ so that the section I defined can be extended in the "obvious way", i.e. "taking derivative at the marked point $y$"? As a working definition of $\overline{M}_{0,1}(\mathbb{P}^2, d)$, I am taking it to be the topological space as defined via Gromov Convergence as defined in the book "J-Holomorphic curves and Symplectic Topology" by McDuff and Salamon.
$\textbf{Remark:}$ I believe the line bundle I am looking for is called the universal (co)tangent bundle, but I am not sure of this. And I am looking for an explicit definition along the lines I have asked.
