I'm trying to learn differential geometry from Isham's book and I want to check if what I'm saying makes sense.
Let $G \rightarrow P \rightarrow \mathcal{M}$ be a principle bundle. Then the connection one-form is the $L(G)$-valued one-form $\omega$ defined as follows. If $\tau \in T_{p}(P)$, then $\omega_{p}(\tau) = i^{-1}(\mathrm{ver}(\tau))$.
Here $i$ is the map from $L(G)$ to the induced vector field on $P$ in the following way: $i(A) = X^{A}$, such that $ X^{A}_{p}(f) = \frac{d}{dt}f(p \exp{tA})|_{t=0}$. Further, $\mathrm{ver}(\tau)$ is the vertical component of $\tau$ from the unique decomposition of $T_{p}(P) = V_{p}(P) \oplus H_{p}(P)$.
Now there are several "inconsistencies" in this definition. The first is that $i: L(G) \rightarrow \mathcal{X}_{ver}(P)$, that is, it maps to the set of vertical vector fields, so that the the inverse map (it exists, since $i$ restricted to this codomain is an isomorphism by construction) maps from vector $\mathbf{fields}$ to lie algebra elements. On the other hand, $\tau$ is a tangent vector, not a field, so this has to be modified a little. The natural way is of course the following: $\mathrm{ver} \tau \in V_{p}(P) \implies \mathrm{ver}{\tau} = X^{A}_{p}$, then $i^{-1}(\tau) \equiv A$.
But then on the other hand, let $\langle \omega, X \rangle = L \in L(G)$ by definition. Then by left invariance, we need $l_{pg^{-1}*}(L_{g}) = L_{p}$, or otherwise $l_{pg^{-1}*}(\langle \omega, X \rangle(g)) = \langle \omega, X \rangle(p)$. But $\langle \omega, X \rangle(p) = \langle \omega_p, X_p \rangle = i^{-1}(\mathrm{ver}X_{p})$, and so we require $l_{pg^{-1}*}(i^{-1}(\mathrm{ver} X_{g})) = i^{-1}(\mathrm{ver}X_{p})$.
This is true because the assignment of vertical subspaces is smooth, and so if $X$ is a vector field then $\mathrm{ver} X$ is also a smooth vector field, and since the set of these is in isomorphism with the $L(G)$, $i^{-1}(\mathrm{ver} X_{g}) = i^{-1}(\mathrm{ver}X_{p})= K$ (say). Then since $K$ is left-invariant, we are done.
Just want to make sure my construction makes sense and is what is usually used. Thanks for any help and comments!