In Tensor Norms and Operator Ideals by Defant and Floret, the authors state in Remark 5.4 that the "natural" mappings $$E\:\hat\otimes_\pi F\to E\:\hat\otimes_\varepsilon\:F\hookrightarrow\mathfrak B(E',F')\tag1$$ $$E\:\hat\otimes_\pi F\to (E'\otimes_\varepsilon F')'\hookrightarrow\mathfrak B(E',F')\tag2$$ are both injective, if one of them is. Above, $E,F$ are Banach spaces, $\hat\otimes_\pi$ denotes the completion wrt projective tensor norm, $\hat\otimes_\varepsilon$ denotes the completion wrt injective tensor norm, $\otimes_\varepsilon$ denote the injective tensor product, $\mathfrak B(E',F')$ denotes the space of bounded bilinear forms on $E'\times F'$ and $\hookrightarrow$ indicates an injection.
I don't get why this is correct. How can we show it?
(Is this just a simple result which is correct for any mappings $A\to B\hookrightarrow C$ and $A\to\tilde B\hookrightarrow C$?)