Remark 11.10.3.6 (Uniqueness). Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of categories. Suppose we are given a pair of morphisms $f: X \rightarrow Y$ and $f': X' \rightarrow Y$ in the category $\operatorname{\mathcal{D}}$ having the same target, with $U(X') = U(X)$ and $U(f') = U(f)$. If the morphism $f$ is locally $U$-cartesian, then the morphism $f'$ factors uniquely as a composition $f' = f \circ e$, where $e: X' \rightarrow X$ is a morphism satisfying $U(e) = \operatorname{id}_{ U(X)}$. In this case, the morphism $f'$ is locally $U$-cartesian if and only if $e$ is an isomorphism.
We can summarize the situation more informally as follows: if $Y$ is an object of the category $\operatorname{\mathcal{D}}$ and $\overline{f}: \overline{X} \rightarrow Y$ is a morphism in $\operatorname{\mathcal{C}}$ which can be lifted to a locally cartesian morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{D}}$, then the morphism $f$ is uniquely determined up to isomorphism.