First I want to clarify what I mean by the Kodaira-Spencer map.

Let's have a family of deformations $\pi:\mathcal{X}\rightarrow B$ of a complex manifold $X=\mathcal{X}_0:=f^{-1}(0)$ (by that I mean that $f$ is an holomorphic proper submersion between the complex manifolds $\mathcal{X}$ and $B$). Then the Kodaira-Spencer map $\rho:T_{B,b}\rightarrow H^1(\mathcal{X}_b,T_{\mathcal{X}_b})$ is the map induced in cohomology by the exact sequence of sheaves $$ 0\rightarrow T_{\mathcal{X}_b}\rightarrow T_{\mathcal{X}|\mathcal{X}_b}\rightarrow f^*T_B|\mathcal{X}_b\rightarrow 0 $$ By "versal deformation" $Def(X)$ I mean that any other deformation $\mathcal{X}\rightarrow B$ of $X$ can be obtained as the pullback deformation for a certain map $g:B\rightarrow Def(X)$ and the differential of $g$ is unique.

My question is: why is the Kodaira-Spencer map for the versal deformation bijective?

This is not clear to me. Also, some authors define the versal deformation as the deformation for which the Kodaira-Spencer map is bijective.