TITLE: Is the product of two objects in a concrete category always a sub-object of the product of these objects viewed as sets?
QUESTION [8 upvotes]: Is the product of two objects in a concrete category always a sub-object of the product of these objects in the category of sets, i.e. is their product in the concrete category a subset of their cartesian product?

REPLY [5 votes]: Let $P : \text{Set}^{op} \to \text{Set}$ be the powerset functor that sends a set $X$ to the set of its subsets and sends a function $f : X \to Y$ to the inverse image function $f^{-1} : PY \to PX$. Since $P$ is faithful we see that $\text{Set}^{op}$ is concrete but also that given any concrete $C$, $C^{op}$ is concrete: if $F : C \to \text{Set}$ is faithful, so is $P \circ F : C^{op} \to \text{Set}$. These $C^{op}$ tend to produce counterexamples (that answer your question negatively) when $C$ is some sort of algebraic category --in which case $C$ itself would not give counterexamples.
To give one specific example, take $C$ to be the category of Abelian groups. So we're making $C^{op}$ concrete by means of the functor $A \mapsto PA=\{$ subsets of $A \}$. The product in $C^{op}$ is the coproduct in $C$ which is the direct sum of Abelian groups. The canonical map $P(A \oplus B) \to PA \times PB$  is just $S \mapsto (S \cap A, S \cap B)$ which is not injective unless $A$ or $B$ is trivial.