Remark 10.3.3.3. In the situation of Definition 10.3.3.1, $Y_0$ is an image of $f$ if and only if there exists a $2$-simplex
10.22
\begin{equation} \begin{gathered}\label{equation:image-in-infinity} \xymatrix@R =50pt@C=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $q$ is a quotient morphism. If this condition is satisfied, we say that the $2$-simplex (10.22) exhibits $Y_0$ as an image of $f$.