Example 10.3.3.5 (Images of Monomorphisms). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \hookrightarrow Y$ be a monomorphism in $\operatorname{\mathcal{C}}$. Since the identity map $\operatorname{id}_{X}$ is a quotient morphism (Exercise 10.3.2.3), the left-degenerate $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{f} & \\ X \ar [ur]^{\operatorname{id}_ X} \ar [rr]^{f} & & Y } \]
exhibits $X$ as an image of $f$.