A property of product category

Question

This property of the product category states that the projections $P$ and $Q$ are "universal" among pairs of functors to $B$ and $C$.

Can someone specify me exactly the sense of that assertion? I have an idea, but it seems to me somewhat trivial.

Answer 1

"Universal" here means that it is "terminal", in the sense that any other triple $(\mathbf D, R, T)$ (I employ your notations) admits a unique morphism, in a suitable category (exercise: define that "suitable category"), to $(\mathbf B \times \mathbf C, P, Q)$. This is precisely the universal property of the product, and defines it uniquely up to a unique isomorphism. The notion of product as given here in the category $\mathbf{Cat}$ of small categories is the general notion of product given in any category. See for example the wikipedia article.

Answer 2

It means that $B \times C$ is terminal among categories equipped with projections on $B$ and $C$: for every category $D$ and every pair of functors $D \to B$, $D \to C$, there exists a unique function $D \to B \times C$ such that the diagram you posted commutes. For more information..