The question comes from Liu's book. Let $f_1 : X_1 \mapsto Y , f_2 : X_2 \mapsto Y$ be morphisms of locally Noetherian schemes of finite type. Let us suppose that for every $y \in Y$ , there exists i = 1 or 2 such that $f_i$ is unramified (resp. étale; resp. smooth) at all points of $f_{i}^{−1} (y)$. I want to show that $X_1 \times_{Y} X_2 \mapsto Y$ is unramified (resp. étale; resp. smooth).
I will wright $p_{1}, p_{2}$ for the canonical projections.
What I've tried for unramified, is to consider an $x \in X_{1} \times_{Y} X_{2}$. We can suppose $f_{1}$ is unramified on the first projection $x'$ of $x$. Let us wright $y := f_{1}(x')$. By Corollary $2.3.$ page $221$, we have $\Omega^{1}_{X_{1}/Y, x'} = 0$ SO that $\Omega^{1}_{X_{1} \times_{Y} X_{2}/X_{2}, x} = p_{1}^{*}(\Omega^{1}_{X_{1}/Y})_{x} = \Omega^{1}_{X_{1}/Y, x'} = 0$ so that $p_{2}$ is unramified in $x$. I can't get more unfortunately...
Étale is the same as unramified and flat. But I'm not able to show the flatness.
For smooth , I take $y \in Y$. We can suppose the points of $f^{-1}_{1}(y)$ are such that $f_{1}$ is smooth on it. This mean, $f_{1}$ is flat on those points and $k(y) \times_{Y} X_{1}$ is smooth as a $k(y)$ variety. So $k(y) \times_{Y} X_{1} \times_{Y} X_{2}$ is also smooth? I don't know if it works.