$\newcommand{\H}{\operatorname{Hom}{}}$ $\newcommand{\HH}{\mathscr{H}}$
Let $f : X \to Y$ a quasi-compact separated morphism of schemes , F a quasi-coherent sheaf on X, $\mathcal{E}$ a locally free sheaf on Y.
In order to prove the projection formula $$f_{*}F\otimes {\mathcal {E}}\to f_{*}(F\otimes f^{*}{\mathcal {E}})$$
I encounter following problem:
Indeed, if I have a concrete map then I can show locally that it's a isomorphism, since locally I could assump $\mathcal{E}= \mathcal{O}_Y ^n$ as free and therefore conclude
$\begin{eqnarray*} f_*(F\otimes f^*E) &\;\cong\;& f_*(F\otimes \mathcal{O}_X^{\,n}) &\;\cong\;& f_*(F\otimes \mathcal{O}_X)^{n} &\;\cong\;& f_*(F)^{n} &\;\cong\;& f_*(F)\otimes \mathcal{O}_Y^{\,n} &\;\cong\;& f_*(F)\otimes E \end{eqnarray*}$
But here occurs my problem:
I need for doing such an argument a concrete morphism $f_{*}F\otimes {\mathcal {E}}\to f_{*}(F\otimes f^{*}{\mathcal {E}})$ which I can't find.
Surely, it suffice to find such one between the presheaves
$U \to f_{*}F(U)\otimes {\mathcal {E}}(U)$
and
$V \to f_{*}(F\otimes f^{*}{\mathcal {E}})(V)$
but I don't find it.
Have already tried adjunction formula without success ...