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Remark 9.3.1.21 (Uniqueness). Let $\mathbb {K}$ be a collection of simplicial sets and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. It follows from Theorem 9.3.1.20 that if there exists a diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}& }$

which exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$, then the inner fibration $\widehat{U}$ is unique up to equivalence (in the sense of Definition 5.1.7.1). More precisely, suppose we are given another commutative diagram

9.32
$$\begin{gathered}\label{equation:uniqueness-of-fiberwise-cocompletion} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H'} & & \widehat{\operatorname{\mathcal{E}}}' \ar [dl]^{ \widehat{U}' } \\ & \operatorname{\mathcal{C}}& } \end{gathered}$$

where $\widehat{U}'$ is a $\mathbb {K}$-cocomplete inner fibration. Theorem 9.3.1.20 then guarantees the existence (and essential uniqueness) of a morphism $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\mathbb {K}}( \widehat{\operatorname{\mathcal{E}}}, \widehat{\operatorname{\mathcal{E}}}' )$ such that $H'$ is isomorphic to $F \circ H$ (as an object of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \widehat{\operatorname{\mathcal{E}}}' )$). In this case, the diagram (9.32) exhibits $\widehat{\operatorname{\mathcal{E}}}'$ as a fiberwise $\mathbb {K}$-cocompletion of $\operatorname{\mathcal{E}}$ if and only if $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{C}}$.