Corollary 10.3.4.7 (04Y3)

Corollary 10.3.4.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ is a universal quotient morphism.

$(2)$

For every pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]

of $\operatorname{\mathcal{C}}$, the morphism $f'$ is a universal quotient morphism.

$(3)$

For every pullback diagram

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^{f'} \ar [r] & X \ar [d]^{f} \\ Y' \ar [r] & Y } \]

of $\operatorname{\mathcal{C}}$, the morphism $f'$ is a quotient morphism.

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 10.3.4.6, the implication $(2) \Rightarrow (3)$ from Remark 10.3.4.2, and the implication $(3) \Rightarrow (1)$ from the criterion of Remark 10.3.1.32 (together with Example 10.3.1.25). $\square$