Homomorphism space/Functorial properties/Fact/Proof/Exercise

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

1. A linear mapping

   $\varphi \colon U\longrightarrow V$

   with 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

   $\psi \colon W\longrightarrow T$

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

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

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