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

   $\phi :U⟶V$

   with another vector space $U$ induces a linear mapping

   ${\mathrm{Hom}}_K(V,W)⟶{\mathrm{Hom}}_K(U,W),f⟼f\circ \phi .$

2. A linear mapping

   $\psi :W⟶T$

   with another vector space $T$ induces a linear mapping

   ${\mathrm{Hom}}_K(V,W)⟶{\mathrm{Hom}}_K(V,T),f⟼\psi \circ f.$

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