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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.

she didn’t have the time or energy to waste on Joe.

While she had her sister in-law as a sister-in-law, she didn’t have her as a mom. There was no one to say, “put on a nice dress, you’ll be meeting the in-laws soon.” Even over a decade later, no one was
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