$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.5.9.16. Let $f: X \rightarrow Y$ be a Kan fibration between Kan complexes, let $n$ be an integer, and let $k$ be a nonnegative integer. If $f$ is $n$-truncated, then the relative diagonal map
\[ \delta : X \rightarrow Y \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, Y) } \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, X) \]
is $(n-k)$-truncated. The converse holds if $k \geq n+2$.
Proof.
We have a commutative diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^-{\delta } \ar [dr]^{f} & & Y \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, Y) } \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, X) \ar [dl] \\ & Y & } \]
where the vertical maps are Kan fibrations. Using Variant 3.5.9.10, we see that $\delta $ is $(n-k)$-truncated if and only if, for each vertex $y \in Y$, the induced map of fibers
\[ X_{y} \rightarrow \{ y\} \times _{ \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, Y) } \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, X) \simeq \operatorname{Fun}( \operatorname{\partial \Delta }^{k}, X_{y} ) \]
is $(n-k)$-truncated. The desired result now follows from Proposition 3.5.9.15.
$\square$