Let $\left\{X_a \right\}_{a \in I}$ be an indexed family of topological spaces. Consider their topological product $X=\prod_{a \in I} X_a$. Let $I'$ be a finite subset of $I$ and define $X' =\prod_{a \in I'} X_a$. Let $y$ be a point of $X$. Define an embedding $j:X' \rightarrow X$ by $j(x')(a)=x'(a)$ if $a \in I'$ and $j(x')(a)=y(a)$ if $a \in I-I'$, where $x' \in X'$. Let $p_a : X \rightarrow X_a$ be the projection on the $a$ factor $x \mapsto x(a)$.
I read the statement "j is continuous, since all maps $p_a j$ are continuous". I can see that $p_a j$ is continuous for any $a \in I$. But why does this continuity imply the continuity of $j$?