= Solution
The adjunction morphism $f^{-1}f_*\mathcal F\to\mathcal F$, together with $f^{-1}\mathcal E\to f^*\mathcal E$, gives
$$
f^{-1}(f_*\mathcal F\otimes_{\mathcal O_Y}\mathcal E)
\longrightarrow
\mathcal F\otimes_{\mathcal O_X}f^*\mathcal E.
$$
Adjunction between the <inverse image sheaf> and <direct image sheaf> turns this into the <projection formula for sheaves> morphism
$$
f_*\mathcal F\otimes_{\mathcal O_Y}\mathcal E
\longrightarrow
f_*\bigl(\mathcal F\otimes_{\mathcal O_X}f^*\mathcal E\bigr).
$$
On local sections it sends a pure tensor $s\otimes e$ over $V\subseteq Y$ to $s\otimes f^\#e$ over $f^{-1}V$.
Whether this morphism is an isomorphism is local on $Y$. If $\mathcal E|_V\cong\mathcal O_V^{\oplus r}$ for finite $r$, its restriction becomes the canonical identification
$$
(f_*\mathcal F|_V)^{\oplus r}
\longrightarrow
f_*\bigl(\mathcal F|_{f^{-1}V}^{\oplus r}\bigr)
=(f_*\mathcal F|_V)^{\oplus r}.
$$
Thus the projection-formula morphism is an isomorphism whenever $\mathcal E$ is <locally free sheaf>[locally free] of finite rank.
Solved by gpt-5.6-sol high.
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