[Jim wrote:](https://forum.azimuthproject.org/discussion/comment/16770/#Comment_16770):

> Are the preceding ones correct? Are the names I've lifted from Wikipedia consistent with the current book?

Your chart is fine, though it breaks my heart that you didn't write it in LaTeX.

There's a bit of variability in how people use names, which I've tried to explain in my posts. A lot of these arise because there are a few terms that everyone learns in math courses which get revamped when people study specialized branches of math connected to category theory - like algebraic geometry - where more intricate things become important:

1) As you note, "preimage" and "inverse image" are synonyms. Both these are standard terms that all mathematicians know. Usually they use the symbol \$$f^{-1}(S)\$$, which I've deprecated for reasons explained in this post: like the term "inverse image", it can fool beginners into thinking the function \$$f\$$ has an inverse \$$f^{-1}\$$ when it doesn't. \$$f^*(S)\$$ is used by category theorists and algebraic geometers.

2) The "proper direct image" \$$f_!(S)\$$ is used by category theorists and algebraic geometers; ordinary mathematicians call this thing merely the "image" and denote it as \$$f(S)\$$. The notation \$$f(S)\$$ is sloppy because strictly speaking we should only apply \$$f : X \to Y\$$ to an element of \$$X\$$, not a subset \$$S \subseteq X\$$.

3) The concept of "direct image" \$$f_*(S)\$$ is not widely known outside algebraic geometry and topos theory, so there's less variability of terminology here.

4) The "proper inverse image" \$$f^!(S)\$$ is even less widely known outside algebraic geometry and topos theory, and you've just pointed out why: in the category of sets it's the exact same thing as the "preimage". It only becomes a distinct concept in more fancy contexts that we haven't met yet, and in fact may never meet in this course.