Vector space/I to K/Contravariance in index set/Exercise

Let ${}K$ be a field, and let ${}S$ and ${}T$ be sets. Show that a mapping

$\psi \colon S\longrightarrow T$

defines a linear mapping

$\operatorname {Map} \,{\left(T,K\right)}\longrightarrow \operatorname {Map} \,{\left(S,K\right)},\varphi \longmapsto \varphi \circ \psi .$

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