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. 

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

Then one can define the Kodaira-Spencer map 

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

simply as 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]) $

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 such argument is based on deformation theory, but it is too intuitive to understand. 

First of all, it is not clear what the target space of $f$ is.

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