Let $(G\_i)\_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod\_{i\in I}G\_i$ denote their [categorical product](https://en.wikipedia.org/wiki/Product_%28category_theory%29). Why do we have the inequality
$$\chi(\prod\_{i\in I}G\_i) \leq \min\{\chi(G\_i):i\in I\},$$ and not the other way round?



Categorical product of graphs and chromatic number
First, let me state that I don't believe this question is suitable for MO - but I'll give an answer anyway.


The categorical product (in any category) of a family of objects $(G\_i)\_{i\in I}$ is characterized by the fact that for every $j\in I$ there is a projection (a morphism that is canonical in some sense, see also the article you linked to) $$\text{pr}\_j: \prod\_{i\in I} G\_i \to G\_j.$$


Now, in the case of graphs, note that whenever there is a graph homomorphism $f: G\to H$ we have $\chi(G) \leq \chi(H)$. The reason is the following: a coloring of $H$ is a graph homomorphism $c: H \to K$ where $K$ is some complete graph, and so $c\circ f: G \to K$ is also a graph homomorphism, which implies directly that $\chi(G) \leq \chi(H)$.


In our particular case, the statement above implies that $\chi( \prod\_{i\in I} G\_i \leq \chi(G\_j)$ for all $j\in I$, which is equivalent to $\chi( \prod\_{i\in I} G\_i \leq \min\{\chi(G\_i):i\in I\}$.


