Hom-functors preserve limits

This is a very useful property of hom-functor.

Let $C$ be a category, then its hom-functor can be wrote as

\text{Hom}_C : C^{op} \times C \to Sets

If the limit $\varprojlim X_i$ exists in $C$ , then for all $Y \in \text{Ob}(C)$ there is a natural isomorphism

\text{Hom}_C(Y, \varprojlim_i X_i) \cong \varprojlim_i (\text{Hom}_C(Y, X_i))

If the colimit $\varinjlim_i X_i$ exists in $C$ , then for all $Y \in \text{Ob}(C)$ there is a natural isomorphism

\text{Hom}_C(\varinjlim_i X_i, Y) \cong \varprojlim_i (\text{Hom}_C(X_i, Y))

See nLab for more details.

Proposition. Hom-functors preserve limits [math-TGEI]