Homomorphism space/Functorial properties/Fact

Let ${}K$ be a field, and let ${}V$ and ${}W$ be ${}K$-vector spaces. Then the following hold.

1. A linear mapping

   $\varphi \colon U\longrightarrow V$

   from another vector space ${}U$ induces a linear mapping

   $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(U,W\right)},f\longmapsto f\circ \varphi .$

2. A linear mapping

   $\psi \colon W\longrightarrow T$

   to another vector space ${}T$ induces a linear mapping

   $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left(V,T\right)},f\longmapsto \psi \circ f.$

Write a proof