ag.algebraic geometry – Obstruction to deformation of composite morphism (Reference request + question)

Let $f_0:X_0xrightarrow{g_0}Y_0xrightarrow{h_0}Z_0/S_0$ be a morphism of smooth projective $S_0$-schemes such that $g_0,h_0$ are flat. Let $S_0subset S$ be a first-order thickening, and let $X,Y,Z$ be $S$-schemes restricting to $X_0,Y_0,Z_0$ over $S_0$. Let $sigma_{f_0}$ (resp. $sigma_{g_0}$ and $sigma_{h_0}$) denote the obstruction to lifting $f_0$ to $f:Xlongrightarrow Z/S$ (resp. $g_0$ to $g:Xlongrightarrow Y/S$ and $h_0$ to $h:Ylongrightarrow Z/S$)

(1) I read in some deformation theory notes that $sigma_{f_0}=g_0^ast sigma_{h_0}$. Is there a reference for this fact?

(2) Assuming that what I wrote in point (1) is correct, suppose furthermore that $S_0$ is the spectrum of an algebraically closed field, that $sigma_{f_0}=0$, and that $g_0$ a finite separable morphism. Is it true that $sigma_{h_0}=0?$