Let $G_1=\langle S_1\mid R_1\rangle,\ G_2=\langle S_2 \mid R_2 \rangle $ and $S_1\cap S_2=\emptyset$.
Show that $G_1\times G_2=\langle S_1\cup S_2\mid R_1\cup R_2\cup R\rangle$ where $R=\{S^{-1}T^{-1}ST=1\},\ S\in S_1,\ T\in S_2$.
Question:
Since $S_1\cup S_2$ isn't a subgroup of the outer direct product, one should probably identify the outer with the inner product for given presentations. Is that correct? If yes, what are the factors of the inner product that is isomorphic to $G_1\times G_2$? I would assume they are the smallest normal subgroups $N_1$ and $N_2$ containing $S_1$ and $S_2$.