What are the morphisms in the category of unramified coverings over a compact Riemann surface?

Question

Fix a compact Riemann surface $S$, and finite a set of branch points $B \subseteq S$. Consider the collection of Riemann surfaces $S_1$ and mermorphic functions $f: S_1 \rightarrow S$, such that $f$ branches over $B$. This gives an unramified covering $S_1 - f^{-1}(B) \rightarrow S- B$. What are the morphisms in this category of unramified coverings?

Answer 1

Well, you could define them to be whatever you want them to be, but lacking any particular contextual clues, I would assume that the morphisms are meant to be the following. If $f:S_1\to S$ and $g:S_2\to S$ are holomorphic maps that ramify only over $B$, then a morphism from $f$ to $g$ is a holomorphic map $h:S_1\to S_2$ such that $gh=f$.