I met this problem in the D.Huybrechts' book Fourier-Mukai transform in algebraic geometry

Let $X$ be a smooth projective variety over an algebraically closed field.

And let $\mathcal F$ be a coherent sheaf on $X \times X$ which is flat over the first factor.

Futhermore, assume $\mathcal F_x$ is concentrated in $x$ for every $x$.

Since $\mathcal F$ is flat over $X$, the Kodaira-Spencer map

$\kappa(x) : Ext^1_X(k(x), k(x)) \rightarrow Ext^1_X(\mathcal F_x, \mathcal F_x)$

is nothing but a Fourier-Mukai transformation $\Phi_{\mathcal F}$

$\Phi_{\mathcal F} : Hom_{D^b(X)} (k(x), k(x)[1]) \rightarrow Hom_{D^b(X)}(\mathcal F_x, \mathcal F_x[1]) $

( For notational convenience, I wrote $\mathcal F_x := \Phi_{\mathcal F}(k(x)) =$ $\mathcal {Li}^*_{x\times X} \mathcal F$ )

At this point, author proved $\kappa(x)$ is injective for the generic point $x \in X$ by the non-rigourous shortcut;

The map $f:x \mapsto \mathcal F_x$ is injective by assumptions. Hence $\kappa(x) := df(x)$ is injective at generic point.

(see D.Huybrechts, Fourier-Mukai transform in algebraic geometry, Ch7, Prop7.1)

Even though it is not even clear where the image of $f$ lives, the argument is quite persuasive and even geometrically intuitive. But I failed to make it rigourous. Could someone explain to me what's going on? Thanks in advance