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.