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Q:

Is it necessary to show the existence of a $T$ that is surjective/injective for $V \subseteq W$

This is a much more general version of Question – Is it necessary to show that every element in the codomain is in the image?
Suppose that $V$ and $W$ are vector spaces. Is it necessary to show that the existence of $T$, which is surjective/injective for $V \subseteq W$, implies that every element in the codomain is in the image?
I think it is necessary, in order to prove the existence of an isomorphism (and even an automorphism in the case of a vector space with $char(k) = 0$).
(Edited) Thanks for the quick answer. My question is about the necessity of proving $T$ surjective/injective. I believe that the proof of existence of an automorphism can be done with an argument of universal property (which is a proof technique) or existence of a natural isomorphism.

A:

You need surjectivity for surjectivity to be defined and for it to be unique. Injectivity isn’t needed for injectivity to exist, but it is essential for surjectivity to exist: there is no guarantee that there is a unique map from one space to another that preserves the inclusion relation.

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