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.

Then we can define the Kodaira-Spencer map at $x \in X$ by the Fourier-Mukai transform $\Phi_{\mathcal F}$.

(for notational convinience, write $\mathcal F_x := \Phi_{\mathcal F}(k(x)) =$ $\mathcal {Li}^*_{x\times X} \mathcal F$)

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

Of course, it is equal to the Kodaira-Spencer map

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

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

In this situation, I want to understand the following argument ;

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)

I think above argument is based on deformation theory, but it is too intuitive to understand.

First of all, what is the target space of $f$ ?

Could anyone explain to me in detail what's going on? Thank you in advance.