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Proposition. Hom-functor preserves limits [math-TGEI]
This is a very useful property of hom-functor.

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

HomC:Cop×C→Sets\text{Hom}_C : C^{op} \times C \to Sets

If the limit lim⁡Xi\lim X_i exists in CC, then for all Y∈Ob(C)Y \in \text{Ob}(C) there is a natural isomorphism

HomC(Y,lim⁡iXi)≃lim⁡i(HomC(Y,Xi))\text{Hom}_C(Y, \lim_i X_i) \simeq \lim_i (\text{Hom}_C(Y, X_i))

If the colimit colimiXi\text{colim}_i X_i exists in CC, then for all Y∈Ob(C)Y \in \text{Ob}(C) there is a natural isomorphism

HomC(colimiXi,Y)≃lim⁡i(HomC(Xi,Y))\text{Hom}_C(\text{colim}_i X_i, Y) \simeq \lim_i (\text{Hom}_C(X_i, Y))

See nLab for more details.