Let $f_0:X_0\xrightarrow{g_0}Y_0\xrightarrow{h_0}Z_0/S_0$ be a morphism of smooth projective $S_0$-schemes such that $g_0,h_0$ are flat. Let $S_0\subset S$ 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 $f:X\longrightarrow Z/S$ (resp. $g_0$ to $g:X\longrightarrow Y/S$ and $h_0$ to $h:Y\longrightarrow Z/S$)
(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 $\sigma_{h_0}=0?$
