binary products in category of simple graphs

Question

I'm trying to wrap my head around what binary products in the category of simple graphs look like. Suppose we have $A, B$ both simple graphs and projection morphisms $p_1 : A \times B \to A$ and $p_2 : A \times B \to B$. Suppose $Y$ is a simple graph and morphisms $f : Y \to A, g : Y \to B$ exist. Then would the product of morphisms $f$ and $g$ just look like $\langle f, g \rangle (x) = (f(x), g(x))$?

Answer 1

Short answer: Yes.

The underlying set of $\mathbf{A}\times\mathbf{B}$ is the Cartesian product $A\times B$, so $h: x\mapsto (f(x),g(x))$ is the only possible candidate for the morphism $h:\mathbf{Y}\rightarrow \mathbf{A}\times\mathbf{B}$ such that $p_1 \circ h = f$ and $p_2 \circ h = g$. (In the language of category theory, the category of graphs is concrete over the category of sets via the forgetful functor, and Cartesian product is the category-theoretical product in $\mathbf{Set}$).

It only remains to check that $h$ is graph homomorphism. But that is easy. Indeed, if you have $u,v\in Y$ such that $u\sim v$, then by assumption $f(u)\sim f(v)$ and $g(u)\sim g(v)$, and hence $h(u)=(f(u),g(u))\sim(f(v),g(v))=h(v)$ in $\mathbf{A}\times\mathbf{B}$. (Just for clarity, for $(a,b),(c,d)\in\mathbf{A}\times\mathbf{B}$ we have $(a,b)\sim (c,d) \iff a\sim c \land b\sim d$).