Corollary 21.3.3. Let \(x\) be an object of \(C\). Then the slice category \(C_{/x}\) admits a terminal object \((x,\id _x)\), while \(C_{x/}\) admits an initial object \((x,\id _x)\).
Proof. The claim for \(C_{x/}\) is immediate from Lemma 21.3.2, since for every \((y, f) \in C_{/x}\) the induced map \(\id _x \circ -\colon \Hom _C(y,x) \to \Hom _C(y,x)\) is an equivalence, hence its fiber over \(f\) is contractible. The claim for \(C_{/x}\) is dual. □
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