Topological spaces form a simplicial category, where the hom-simplicial set between the spaces $X,Y$ is roughly $$\hom(X,Y)_n =\text{Top}(|\Delta^n|\times X,Y)$$
Show that $\text{Top}$ is locally Kan, i.e. that the $\hom(X,Y)_\bullet$ are Kan complexes.
If we restrict to a convenient category of spaces, say compactly generated, then we have an adjunction $$ \text{Top}(Z\times X,Y)\cong \text{Top}(Z,\text{Map}(X,Y)). $$
So that $$\hom(X,Y)_n\cong \text{Top}(|\Delta^n|,\text{Map}(X,Y)) \cong \text{Sing}(\text{Map}(X,Y))_n.$$
We figure that $\hom(X,Y)_\bullet$ is a Kan complex because $\text{Sing}(\text{Map}(X,Y))_\bullet$ always is.
How to show that the simplicial enrichment of the category $\text{Top}$ of all spaces is a simplicial category?
This is Exercise 2.4.1.10 at Kerodon.