In his proof of proposition 2.5.6 on pg. 113 Hartshorne seems to use the term lift in two different ways. Fist he seems to use it in the context of surjectivity of a map of sheaves: In the sense that $s$ is locally the image of sections of $\mathcal{F}$. Then immediately after he uses it to describe a section 'lifting' to a global section on the same sheaf. In both cases it seems to describe a section lying in the set theoretic image of a map, but in the first case the map is a map of sheaves and in the other its a restriction map. Is this use of terminology correct or consistent throughout the book?
Edit: Further in the proof he uses the term extends' to destine a section lifting to a section on a larger open set of the same sheaf. This seems to indicate his usage is in fact inconsistent