# Inverse Image Sheaf and pullback of a Vector Bundle.

Let $$M,N$$ be smooth manifolds and $$f:N\to M$$ a smooth map. Denote by $$\mathcal{O}_M,\mathcal{O}_N$$ the corresponding sheaves of smooth functions. If we regard $$\mathcal{O}_M$$ as the sheaf of smooth sections of the trivial bundle $$M\times \mathbb{R}$$ over $$M,$$ does the sheaf of sections of the pullback bundle coincide with the pullback sheaf $$f^*\mathcal{O}_M.$$ More generally does this happen for any vector bundle? I think it does, although wikipedia states that in general the sheaf of sections of the pullback bundle only coincides with the inverse image sheaf. Nonetheless, since pullback of a trivial vector bundle is a trivial vector bundle. Then the sheaf of sections of the pullback of $$M\times \mathbb{R}$$ must coincide with $$\mathcal{O}_N.$$ But this seems weird. Edit: I made a mistake in the statement the map is supposed to go like $$f:N \to M.$$