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Stream: learning: questions

Topic: Exact squares with an additional property

fosco (Jul 19 2023 at 17:58):

Every arrangement of adjoints with $i,q_R$ fully faithful functors

image.png

induces a square filled by a 2-cell

image.png

obtained with the counit and unit of the adjoints. This squares evidently satisfies the Beck-Chevalley condition or, with a slight abuse of notation, it is "exact".

If, now, $A,B,C$ are pointed categories, or additive ones, it makes sense to ask the additional property that the square

image.png

written diagrammatically, is a pullback (in the category of endofunctors of $B$). How can this condition be stated, in terms of the 2-dimensional pasting diagram above?