Endomorphism space/K^2/Squaring/Exercise

We consider the space of endomorphisms ${}\operatorname {End} _{}{\left(K^{2}\right)}=\operatorname {Hom} _{K}{\left(K^{2},K^{2}\right)}$. Is the mapping

$\operatorname {End} _{}{\left(K^{2}\right)}\longrightarrow \operatorname {End} _{}{\left(K^{2}\right)},\varphi \longmapsto \varphi \circ \varphi ,$

a linear mapping?

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