Suppose $X$ and $Y$ are both regular schemes and the morphism $X \to Y$ is a local complete intersection (meaning it is a composition of a regular immersion and a smooth morphism, but in general not flat, e.g. $\mathbb{A}^2$ blowup the origin). Are there any conditions which could ensure that the morphism $X \times_Y X \to X$ (second projection) is a local complete intersection again?
