Hom animae

Definition 1.4.17 (Hom anima). Let \(C\) be any \(\infty \)-category. Given objects \(x\) and \(y\) of \(C\), we define the hom anima \(\Hom _C(x,y)\) via the following pullback square:

Observe that the objects of \(\Hom _C(x,y)\) are triples \((f,\alpha ,\beta )\), where \(f\colon x' \to y'\) is a morphism in \(C\) and \(\alpha \colon x \cong x'\) and \(\beta \colon y \cong y'\) are isomorphisms in \(C\). We will often abuse notation and pretend that \(x = x'\) and \(y = y'\).