Suppose $X$ is a Hausdorff space (no other assumptions on it) and denote by $\eta:X\to \beta X$ the canonical map to its Stone–Čech compactification (compactification). Consider a continuous function $f:A\times X\to B$ where both $A$ and $B$ are compact Hausdorff. Is it true that there exist an extension $\beta (f): A\times \beta X\to B$ (even not unique), such that $f=\beta(f)\circ (id_A\times \eta)$ ?
I know that if $A$ is the one point space the answer is yes, as it reduces to the standard universal property of the Stone–Čech compactification. I wonder if this can be generalized for any compact Hausdorff space. Does $\beta$ commute with finite products in this setting? It would be sufficient.