Equivalent definitions of connection on a vector field Let $X$ be a nice variety resp. a manifold over the complex numbers. One defines a connection on a vector bundle $V$ on over $X$ as a $\mathbb C-$linear sheaf homomorphism $\nabla : V\rightarrow V\otimes \Omega^1$ which satisfies the Leibniz rule. I have read that this is equivalent to giving for each local vector field $Y\in Der_{\mathbb C}(\mathcal O_X)$ a $\mathbb C-$ linear sheaf homomorphism $\nabla_Y : V \rightarrow V$ with (1) Leibniz rule (2) $\nabla_{fY+gZ}=f\nabla_Y+g\nabla_Z$ for $f,g \in \mathcal O_X$ and $Y,Z$ local vector fields. I can prove that each connection in the first sense implies a connection in the second sense. But I don't see how you get from the datum of thhe $\nabla_Y$ a connection in the first sense. Remark: By a vector field I understand a linear derivation of the structure sheaf into itself.

Using the fact that $\Omega^1$ is $\mathcal{O}_X$-locally free and dual to the sheaf of vector fields you should be able to prove $Hom_{\mathcal{O}_X}(\Theta_X,\mathcal{E}nd_\mathbb{C}(V)) = Hom_\mathbb{C}(V,V\otimes_{\mathcal{O}_X} \Omega^1)$. In local complex coordinates $(x_1,\ldots,x_n)$, $\\{\
abla_Y\\}$ is mapped to $\
abla: V \to V\otimes \Omega^1$ defined by $$ \
abla(v) = \sum_{i=1}^n \
abla_{\partial_i}(v) \otimes dx_i $$ (This the analogue of $df = \sum_i \frac{\partial f}{\partial x_i} dx_i$). Condition (2) ensures that this formula does not depend on the choice of coordinates.