I am reading Claudio Procesi's book on Lie groups and on page 105 there is something I don't understand.
Let $U,V,W$ be vector spaces. Let us consider the product space $\hom(V,W) \times \hom(U,V) $ together with the map
$$f : \hom(V,W) \times \hom(U,V) \rightarrow \hom(U,W)$$
that sends a pair $(v, \varphi)$ to $v \circ \varphi$. Suppose I have an elementary tensor $v \otimes \varphi$ in $ \hom(V,W) \otimes\hom(U,V) $. By the universal property of tensor products there is a unique linear map
$$ L : \hom(V,W) \otimes \hom(U,V) \rightarrow \hom(U,W)$$
such that on elementary tensors, $L(v\otimes \varphi) = v \circ \varphi$.
Now 1.5.1 of pg 105 claims that for any $u \in U$, we have that
$$(v \otimes \varphi)(u) = v \langle \varphi | u\rangle$$
where $\langle \varphi | u\rangle$ is defined on page 16 to be the value of the linear form $\varphi$ on $u$. I am confused as to how they obtained that equality above - is it really an equality? I don't see it directly but I think it comes from the fact that
$$\hom(U,V) \cong V \otimes U^\ast.$$