Sheaves and bundle morphisms I'm currently trying to reunderstand differential geometry using shaves. Let $f:(E_1,M,\pi_1)\rightarrow (E_2,M,\pi_2)$ be a $\mathcal{C}^{k}$ vector bundle morphism, where $M, E_1$ and $E_2$ are a $\mathcal{C}^{k}$ Hausdorff second countable manifolds. Consider the presheaf of modules of sections $\mathcal{F}^{k}(U):=\{s:U\subset M\rightarrow \operatorname{Im}(f)\subset E_2:\pi_2\circ s=Id_{U}\textrm{ and }s\textrm{ is } \mathcal{C}^{k}\textrm{ differentiable}\}$. Is this presheaf a sheaf?
Books and other sources are also welcome.
Thanks in advance.
 A: You have to check that it satisfies the locality and gluing properties, which I think is a mere computation:
(Locality) Suppose $U$ is an open set, $\{U_i\}_{i\in I}$ is an open cover of U $U$, and $s,t \in \mathcal{F}(U)$ are sections. If $s|_{U_i}=t|_{U_i}$ for all $i\in I$, then $s=t$. If not, i.e, if there exists $x$ such that $s(x)\neq t(x)$ for $x\in U_{i_0}\subset U$ then $s|_{U_{i_0}}(x)\neq t|_{U_{i_0}}(x)$, which is a contradiction.
(Gluing) Suppose $U$ is an open set, $\{U_i\}_{i\in I}$ is an open cover of $U$, and $\{s_i\in \mathcal{F}(U_i)\}$ is a family of sections. Suppose all pairs of sections agree on the overlap of their domains, that is, if $s_i|_{U_{ij}}=s_j|_{U_{ij}}$ for all $i,j\in I$.
You can define $s:U\to E_2$ by
$$
s(x)=s_i(x) \text{ for } x\in U_i.
$$
It is well defined, since if $x\in U_{ij}$ then $s_i(x)=s_j(x)$.
It is trivial to check that $s\in \mathcal{F}(U)$, so you have that there exists a section $s\in \mathcal{F}(U)$ such that $s|_{U_i}=s_i$ for all $i\in I$, as required.

In general, vector bundles are locally free sheaves. The kernel and image of vector bundle morphisms are not vector bundles (unless they are constant rank morphisms) but they are, at least, sheaves. Those sheaves are therefore not necessarily locally free ones but coherent sheaves. Coherent sheaves constitute the class of sheaves such that their image and kernel are still sheaves of the same class. You can understand them as a generalization of vector bundles.
I studied this stuff so many years ago, I hope I am not missing something...
