I have been looking at this document: https://math.mit.edu/classes/18.783/2015/LectureNotes5.pdf.
In the proof of Lemma 5.21, it is expressed that an isogeny can be supposed to be a rational map on projective coordinates. I don't see how this is simply a supposition. Wouldn't the proof of the lemma require some detail as to why all isogenies can be expressed in this way.
I am also studying from Silverman, and that text also makes no specific remark as to why the standard form for isogenies on elliptic curves are rational maps. Am I missing a very simple contradiction argument as to why an isogeny must be a rational map?