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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.
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?
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