Let $V$ be the vertical projection of an Ehresmann connection on a submersion $X\to Y$. Write $\mathrm T(X/Y)$ for the vertical bundle. Then the vertical projection is a vector bundle morphism retracting the inclusion $\mathrm T(X/Y)\to \mathrm T(X)$.
By relativizing the natural map $U^\vee\otimes V\to \mathsf{Vect}(U,V)$ (an isomorphism in the finite dimensional case) over a manifold we get an isomorphism between the sections of the vector bundle $\mathrm T^\vee X\otimes \mathrm T(X/Y)$ and vector bundle morphisms $\mathrm T(X)\to \mathrm T(X/Y)$.
A section of the latter is precisely a 1-form on $X$ valued in the vertical bundle, so the latter isomorphism uniquely associates a connection 1-form to an Ehresmann connection.
I would like to distinguish which of these 1-forms is a connection 1-form. Since the vertical projection is characterized by retracting the inclusion of the vertical bundle into $\mathrm T(X)$, the task is to translate this retraction condition into a condition on the associated connection 1-form.
Unfortunately, I don't see how to translate the geometric condition of retracting the inclusion into a condition on a tensor valued section of a bundle. I think it should be very simple (since it relies only on the natural map above), but I can't put my finger on it.
Question. What condition on a 1-form valued in the vertical bundle characterizes it as a connection 1-form?