I'm stuck at two places in these lecture notes:
1) Consider on pp 40:
What does the last equation mean? This makes no sense to me. $\varphi_i$, $i=1,2$ can't denote the $i$th coordinate function, since that takes 2 arguments, not one. What does it the denote? And what is the $\cdot$? The scalar product?!
2) Consider on pp. 53
($T^r_s(E)$ are the $\binom{r}{s}$-tensors on the vector space $E$, as defined previously in the notes - see the link.) I cannot understand the equation $e(\varphi^* (\beta)) = \varphi (e) (\beta)$, or why $\varphi_0^1$ is in some sense isomorphic to $\varphi$.
Here $\varphi: E \rightarrow F$ for finite-dim. real vector spaces $E,F$ and $\varphi^*$ is the adjoint, $e \in E^{**}$ and $\beta \in F^*$ (this is all explained in the lines above this example).
Actually the equation makes no sense in my opinion, since $\varphi(e)$ is an element of $F$, so it makes no sense to apply $\beta$ here.
As to your first question, the map $\varphi$ has two components $$(u, f)\longmapsto \varphi_1(u)\quad \textrm{and}\quad (u, f)\longmapsto \varphi_2(u)\cdot f,$$
where $\varphi_1:U\longrightarrow F$ and $\varphi_2:U\longrightarrow \mathsf{Hom}(F, F^\prime)$ where $\mathsf{Hom}(F, F^\prime)$ is the vector space of linear maps from $F$ to $F^\prime$. This applies to the definition of local vector bundle homomorphism.
If you change "homomorphism" by "isomorphism" you must change $\mathsf{Hom}(F, F^\prime)$ by $\mathsf{Iso}(F, F^\prime)$ where $\mathsf{Iso}(F, F^\prime)$ is the vector space of linear isomorphism from $F$ to $F^\prime$.
As to your second question, when he writes $\varphi_2(u)\cdot f$ it means you are evaluating the linear map $\varphi_2(u):F\longrightarrow F^\prime$ in the point $f\in F$.
As to your last question, I believe he is defining $\varphi_0^1:E\longrightarrow F$ via duality. There are natural pairings $$\langle \cdot , \cdot\rangle_E:E\times E^*\longrightarrow \mathbb R$$ and $$\langle \cdot, \cdot\rangle_F: F\times F^*\longrightarrow \mathbb R$$ and he defines $\varphi_0^1:E\longrightarrow F$ as the unique map such that $$\langle \varphi_0^1(e), \beta\rangle_F=\langle e, \varphi^*(\beta)\rangle_E=\langle \varphi(e), \beta\rangle$$ for every $e\in E$ and $\beta\in F^*$.