Let $I$ be an index set, $X_i$ be a topological space for each $i \in I$ and $X = \prod_{i \in I} X_i$ the product of all $X_i$. Then the product topology is exactly the initial topology with respect to the canonical projections $\pi_i: X \to X_i$.
Let $A$ be a topological subspace of $X$ (i.e. $A$ has the subspace topology in $X$) and $\varphi_i$ are the restriciton of the projections $\pi_i$ on $A$.
Question: Is $A$ the initial topology with respect to the maps $\varphi_i$?
I think that the arguments which I need can be found in this nice explanation of Henno Brandsma, but I am unable to convert it into a line of reasoning which convinces me. Could you please answer my question and give an explanation? Thank you!