  In this paper we discuss the relationship between direct products of
monounary algebras and their components, with respect to the properties of
residual finiteness, strong/weak subalgebra separability, and complete
separability. For each of these properties $\mathcal{P}$, we give a graphical
criterion $\mathcal{C_P}$ such that a monounary algebra $A$ has property
$\mathcal{P}$ if and only if it satisfies $\mathcal{C_P}$. We also show that
for a direct product $A\times B$ of monounary algebras, $A\times B$ has
property $\mathcal{P}$ if and only if one of the following is true: either both
$A$ and $B$ have property $\mathcal{P}$, or at least one of $A$ or $B$ are
backwards-bounded, a special property which dominates direct products and which
guarantees all $\mathcal{P}$ hold.
