Following the last piece of the accepted answer to this question, we have the following.
Let $f : M \to N $ be a smooth function between smooth manifolds, $x^i$ be local coordinates on $M$, $\\{y^\alpha \\}$ local coordinates on $N$, $E \to N$ a vector bundle with local frame $\\{ E_A \\}$, then the pullback bundle $f^* E$ has a local frame given by $\\{ E_A \circ f \\}$. Putting things together, we get the following relationship between the connection coefficients for $\nabla$ on $E$ and $f^* \nabla$ on $f^* E$: $$^{f^* \nabla} \Gamma_{Ai}^{B} = {}^{\nabla}\Gamma_{A \alpha}^{B} (df)_{i}^{\alpha}. $$
What confuses me is the following (correct me if I'm wrong). The connection coefficients $^{f^* \nabla} \Gamma_{Ai}^{B}$ are smooth functions from $U \to \mathbb{R}$ for some open set $U \subset M$, the connection coefficients ${}^{\nabla}\Gamma_{A \alpha}^{B}$ are smooth functions from $V \to \mathbb{R}$ for some open set $V \subset N$, but what's happening with $(df)_{i}^{\alpha}$?
From what I understand, $df$ is clearly the differential of $f$, a map from $TM \to TN$, and $(df)_ {i}^{\alpha}$ are, locally, the components of the Jacobian matrix. But for the above equation to work out, the domain of $(df)_{i}^{\alpha}$ must be $U$ and the codomain must be $V$ if we're composing the two terms. I cannot understand how this makes sense, unless I'm missing the obvious, in particular I do not know what the components $(df)_{i}^{\alpha}$ of $df$ actually are, and what their domain and codomain are.