Vector spaces and endomorphisms as $K[X]$-Modules

Question

Given a $K$-vector space $V$ together with a $K$-linear endomorphism $\varphi$. $V$ gets a $K[X]$-module structure on $V$ via $fv:=f(\varphi)(v)$. I would like to show the associative law for the module, but I'm not quite sure how it works. It should be like the following: Let $f,g\in K[X]$. Then $(fg)v=\sum_{ij} (a_i \varphi^i b_j \varphi^j)(v)=\sum_i a_i \varphi^i(\sum_j (b_j \varphi^j)(v))= f(\varphi(g(\varphi(v)))=f(gv)$. Does this make any sense and how is the second equation justified? Thanks a lot!