added (here) also a mentioning of the Beck-Chevalley condition satisied by forming pre-/images

(essentially copied over from what I just added there)

]]>I have slightly adjusted formatting and wording of the remark on inverse image being right adjoint to direct image (here)

]]>For what it’s worth, the statement in question (here) is also stated as Exercise A.4(k) in Lee 2000.

]]>Just to highlight that the question is about the “projection formula” here which was apparently added by Todd Trimble in rev 10 (leaving the direction under question now as “to be continued”).

]]>https://ncatlab.org/nlab/revision/diff/interactions+of+images+and+pre-images+with+unions+and+intersections/17

Now everything there should be OK.

I love nLab! Thanks for your great work! ]]>

Let me see… It looks okay to me: The intersection of images may be larger than the image of the intersection.

In an extreme case, take $S_1$ and $S_2$ to two disjoint copies of the same non-empty abstract set, and each with the same image under $f$. Then the image of the intersection is empty (because the intersection itself is), but the intersection of the images is the image of either subset.

Hm, now I see that this same point is also amplified in a comment right below.

So maybe I am missing your point?

To make sure we are looking at the same part of the entry: You can copy and paste the relevant code from the entry to the Forum here. (Just make sure you check the radio button “Markdown+Itex” below the edit pane here.)

]]>Should "contained in" be removed in the explanation of the properties 2. of Proposition 2.1? ]]>

At *interactions of images and pre-images with unions and intersections* I have added pointer to Lawvere 69 and there at *Adjointness in Foundations* I added a bit more text and cross-references.