Lemma 21.3.2. Consider an object \(x\) of \(C\) and consider two objects \((y,f)\) and \((y',f')\) of the slice category \(C_{/x}\). Then the hom anima in \(C_{/x}\) sits in a pullback square of the form
A dual formula holds for the hom animae of \(C_{x/}\).
Proof. By Chapterexercise 1.4, the hom anima in \(C_{/x}\) may be computed as the following pullback:
Using the equivalence \(\Fun ([1] \times [1],C) \simeq \Fun ([2],C) \times _{\Ar (C)} \Fun ([2],C)\) from the commutative square axiom, the top right hom anima in \(\Ar (C)\) sits in a pullback square as follows:
The claim thus follows by pasting pullback squares. □
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