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Proposition Let $f: X \rightarrow Y$ be a morphism of simplicial sets, and let $Z$ be a Kan complex. If $f$ is a weak homotopy equivalence, then composition with $f$ induces a homotopy equivalence $\operatorname{Fun}( Y, Z) \rightarrow \operatorname{Fun}(X,Z)$.

Proof. By virtue of Remark, it will suffice to show that for every simplicial set $A$, the induced map $\theta : \operatorname{Fun}( A, \operatorname{Fun}(Y, Z) ) \rightarrow \operatorname{Fun}( A, \operatorname{Fun}(X,Z) )$ induces a bijection on connected components. This follows by observing that $\theta $ can be identfied with the map $\operatorname{Fun}(Y, \operatorname{Fun}(A,Z) ) \rightarrow \operatorname{Fun}( X, \operatorname{Fun}(A,Z) )$ given be precomposition with $f$ (since Corollary guarantees that the simplicial set $\operatorname{Fun}(A,Z)$ is a Kan complex). $\square$