Tensor product/Functoriality in vector space/Fact

Let ${}K$ be a field, and let ${}U,V,Z$ denote vector spaces over ${}K$. Then the following statements hold.

1. For a ${}K$-linear mapping ${}\varphi \colon U\rightarrow V$, there exists a natural ${}K$-linear mapping ${}\varphi \otimes _{K}\operatorname {Id} _{Z}\colon U\otimes _{K}Z\rightarrow V\otimes _{K}Z$.
2. If ${}\varphi$ is surjective, then also ${}\varphi \otimes _{K}\operatorname {Id} _{Z}$ is surjective.
3. If is ${}\varphi$ injective, then also ${}\varphi \otimes _{K}\operatorname {Id} _{Z}$ is injective.

Proof, Write another proof