In the book Connections in classical and quantum field theory by L. Mangiarotti, G. Sardanashvily I have a problem with $(1.1.3)$ formula below: $$(x,v)\mapsto (x,\rho_{\xi\zeta}(x,v));$$
namely that $$\rho_{\xi\zeta}(x,v)$$ should lie in $V$ but as is, it lies in $$(U_\xi\cap U_\zeta)\times V$$
What is the correct formula for the 2nd line in $(1.1.3)$ ?
Since $\rho$ acts trivially on the $x$-coordinate, it seems like an abuse of notation that the second coordinate of $\rho$ is $\rho$ itself. You could make up your own notation for the second coordinate such as $\rho(x,v) = (x, \sigma(x,v))$.
In the case of the tangent bundle, if $x^i$ and $y^j$ are coordinates that are overlapping at some point $x$, then the map $\rho$ near $x$ in coordinates would be $\rho(x,v) = (x, M(x)v)$, where $M(x)$ is the linear map sending $\frac{\partial}{\partial x^i}|_x$ to $\frac{\partial y^j}{\partial x^i}(x)\frac{\partial}{\partial y^j}|_x$, and $\sigma(x,v) = M(x)v$. Here $M(x) = \frac{\partial y^j}{\partial x^i}(x)$ is the Jacobian change of basis matrix computed at $x$.