Consider the unit disk $\mathbb D \subset \mathbb R^2$, and a holomorphic map $f :\mathbb D \to \mathbb D$. $df$ can be thought of as $df: T^{0,1}D \to f^*(T^{0,1}D)$(the pull back), that is, an element in $L(TD, T^{\ast}D)$ which is isomorphic to $T^{\ast} D \otimes f^{\ast}TD$. Then $df = f' dz \otimes \frac{\partial}{\partial w}$ in basis.
I was told that there is a "connection" $\nabla$ on $T^{\ast}D \otimes f^{\ast}D$, which is compatible with the hyperbolic metric, and I need to compute $\nabla f'dz \otimes \frac{\partial}{\partial w}$. This boils down to computing $\nabla dz \otimes \frac{\partial}{\partial w}$.
My guess here is that we are viewing $T^{\ast} D \otimes f^{\ast}D$ as a complex vector bundle, then we can define some sort of complex connection on it that will be compatible with the hyperbolic metric on the unit disk. How to define such a connection and what is the compatibility condition?
Update: Let $E$ be a holomorphic line bundle. Then an hermitian structure $H$ on $E$ is given by a positive real function and the Chern connection $E$ is locally given as $\nabla = d + \partial log H$.
How can I use this statement to compute $\nabla dz \otimes \frac{\partial}{\partial w}.
My interpretation is that $\nabla dz \otimes \frac{\partial}{\partial w} = \nabla dz \otimes \frac{\partial}{\partial w} + dz \otimes \nabla \frac{\partial}{\partial w} = \partial log H \otimes \partial w + dz \otimes \frac{\partial}{\partial w}$, what is $d\frac{\partial}{\partial w}$?