The underlying set of the product $X \times Y$ of two schemes is by no means the set-theoretic product of the underlying sets of $X$ and $Y$.
Although I am happy with the abstract definition of fiber products of schemes, I'm not confident with some very basic questions one might ask. One I am specifically thinking about has to do with the connected components of a fiber product.
Say $X$ and $Y$ are schemes over $S$ and let's suppose that $Y$ is connected and that $X = \coprod_{i \in I} X_i$ is the decomposition of $X$ into connected components. Is the connected component decomposition of $X \times_S Y$ simply $\coprod_{i \in I} X_i \times_S Y$?
If so, how can we see this? What about if we replace "connected" with "irreducible".
Thank you for your consideration.