Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 115 / 3 / d / Solution

A covariant derivative is an $\mathbb{R}$-linear mapsatisfying $d_A(fs)=df\otimes s+f{d}_{A}s$. It is local: its value over an open set depends only on the restriction of the section there. A horizontal connection differentiates a section, projects its derivative vertically, and identifies the vertical tangent with the fibre.

In a local frame write $d_{A}s=ds+A_{i}sd{y}^i$. If $x^a$ are coordinates on $M$, the defining identity in the question forceswithThus $A^{\prime }=f^{\ast }A$, the pullback connection.