[m00n]

A map between algebraic curves is defined by polynomials. That the map is defined over K means you can find a coordinate system such that the equations of the curves and the equations of the morphisms have coefficients in K and not some larger ring, eg the complex numbers or a large extension field of F_p(field of p elements)

[ur-whale]

> A map between algebraic curves is defined by polynomials.

That's a statement that requires IMO justification.

Is that really always the case?

Does it somehow stem from the fact that the source and destination sets are - albeit large - finite?

Again: not obvious.

[mti]

A map of algebraic curves is not a special case of a map between two sets. There aren't really source and destination sets to speak of. The fact that it is given by polynomials really is a definition.

The moral "justification" of the definition is something like "algebraic geometry is precisely the study of such objects" or "we want definitions that are stable under ring base change, and this implies polynomials", etc., but we don't formally need to justify definitions.

[One could take a different route to defining those things, in which this becomes a theorem instead of a definition. For example one can define algebraic curves over a field k as contravariant functors from k-algebras to sets satisfying certain additional properties, and then maps of algebraic curves are natural transformations between those functors. The fact that they are given by polynomial equations is then a theorem. Just stating the "additional properties" for a curve is a rather daunting task, though, unfortunately.]