Let $A$, $X$, and $Y$ be arbitrary topological spaces. Let $F:X\times A\rightarrow Y$ be a continuous function. Let $P$ be the map from $X\times A$ to $Y\times A$ taking $(x,a)$ to $(F(x,a),a)$. Does it follow from continuity of $F$ that $P$ is continuous?
Yes. This follows from the fact that a function $U \to V \times W$ is continuous if and only if its component functions $U \to V, U \to W$ are, and from the fact that the projection maps $V \times W \to V$ and $V \times W \to W$ are continuous. Both of these facts in turn follow from the universal property of the product topology.