Kerodon

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Proposition 7.1.4.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$ with the property that $U(f)$ is an isomorphism. Then any two of the following three conditions imply the third:

$(1)$

The object $X$ is $U$-initial.

$(2)$

The object $Y$ is $U$-initial.

$(3)$

The morphism $f$ is an isomorphism.

Proof. Fix an object $Z \in \operatorname{\mathcal{C}}$. We claim that any two of the following three conditions imply the third:

$(1_{Z})$

The functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(X), U(Z) )$.

$(2_{Z})$

The functor $U$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( U(Y), U(Z) )$.

$(3_{Z})$

Precomposition $[f]$ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$ (see Notation 4.6.8.15).

This follows from the commutativity of the diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \ar [r]^-{ \circ [f] } \ar [d] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(Y), U(Z) ) \ar [r]^-{ \circ [ U(f) ] } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}( U(X), U(Z) ) }$

in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$, since the bottom horizontal map is a homotopy equivalence (by virtue of our assumption that $U(f)$ is an isomorphism). Proposition 7.1.4.10 follows by allowing the object $Z$ to vary. $\square$