Given a non-constant isogeny $f : E_1 \rightarrow E_2$ of degree $n$ between elliptic curves, I'm under the impression that there always exists a unique isogeny $g : E_2 \rightarrow E_1$ satisfying $$g \circ f = [n]_{E_1}, \qquad f \circ g = [n]_{E_2},$$ which is called the dual isogeny of $f$. This seems very amazing and mysterious, so:
How does one actually compute the dual isogeny, assuming the elliptic curves under question are described by Weierstrass equations?