Let $\mathbf{C}$ be the class of algebraic structures of some fixed type satisfying some sentence $\phi$. Birkhoff's HSP theorem says that $\mathbf{C}$ is closed under homomorphisms, subalgebras and products if and only if $\phi$ is equivalent to some universally quantified conjuction of equations. In the following article -

Properties preserved under algebraic construction. Author: R. C. Lyndon. Journal: Bull. Amer. Math. Soc. 65 (1959), 287-299. http://www.ams.org/journals/bull/1959-65-05/S0002-9904-1959-10321-9/

Lyndon refers to an "obvious" HD theorem at the top of p293 (as opposed to Birkhoff's UHD(=HSP) theorem) which I think means that the same result holds if $\mathbf{C}$ is only closed under homomorphisms and products, except that we now allow existential as well as universal quantifiers. Let us say we weaken the assumption on $\mathbf{C}$ just a little more, and assume that the homomorphisms considered only come from coordinate functions on products, i.e. what if we only assume that $\mathbf{C}$ satisfies

$X\times Y\in\mathbf{C}\quad\Leftrightarrow\quad X\in\mathbf{C}\textrm{ and }Y\in\mathbf{C}$

(and the same for infinite products, although I think this already follows from the above finite product assumption). What kind of sentence must $\phi$ now be equivalent to?