Remark 10.3.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which has images and admits finite limits. Then, for every pullback diagram
\[ \xymatrix@C =50pt@R=50pt{ X' \ar [r]^-{f'} \ar [d] & Y' \ar [d]^{u} \\ X \ar [r]^-{f} & Y, } \]
we always have a containment $\operatorname{im}(f') \subseteq u^{-1}( \operatorname{im}(f) )$ in $\operatorname{Sub}(Y')$; this follows from the characterization of $\operatorname{im}(f)$ supplied by Proposition 10.3.3.11.