Warning 7.6.5.6. In the situation of Remark 7.6.5.5, the map
\[ \theta : \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, \varprojlim (X) ) \rightarrow \varprojlim ( \operatorname{Hom}_{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}}( Y, X(n) ) ) \]
need not be injective. That is, the forgetful functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ generally does not preserve sequential limits (or colimits).