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.
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