# A natural isomorphism $\mathsf C(\operatorname{colim} F, Y) \cong \lim \mathsf C(F-, Y)$. Understanding $\lim$ as a functor?

In the text, "Topology: A Categorical Approach", there is an exercise generalizing the identity $$\mathsf C(\coprod X_i, Y) \cong \prod \mathsf C(X_i, Y)$$:

Prove that a functor $$F: \mathsf B \to \mathsf C$$ has a colimit if and only if for all objects $$Y \in \mathsf C$$ there is a natural isomorphism: $$\mathsf C(\operatorname{colim} F, Y) \cong \lim \mathsf C(F-, Y).$$

A similar statement and its dual can also be found here: https://ncatlab.org/nlab/show/hom-functor+preserves+limits.

Suppose the colimit of $$F$$ exists. I can understand why these are isomorphic as sets, and this much is proven in the nlab page, but I am not quite sure how to interpret "natural isomorphism" here. I think it is meant to be understood that the isomorphism is natural in $$Y$$. In this case, the left hand side becomes a regular hom-functor, $$\mathsf C(\operatorname{colim} F, -)$$, but if this is correct, then how would one interpret the right hand side? It seems likely that for objects, we have $$Y \mapsto \lim \mathsf C(F-, Y)$$, but I have not been able figure out how such a functor should act on morphisms. Does anyone have any insight on this?

Let $$f : Y_1 \to Y_2$$ be a morphism in $$\mathsf C$$. Considering the object $$\lim \mathsf C(F-,Y_1)$$ as a constant functor, there is a natural transformation $$\lim \mathsf C(F-,Y_1) \longrightarrow \mathsf C(F-,Y_1) \stackrel{f_*}\longrightarrow \mathsf C(F-,Y_2)$$ (the first one is precisely the cone from $$\lim \mathsf C(F-,Y_1)$$ to $$\mathsf C(F-,Y_1)$$, and the second one is the natural transformation whose components are pre-composition by $$f$$) and so, the universal property of $$\lim \mathsf C(F-,Y_2)$$ tells us that there is a unique morphism $$f_\flat : \lim \mathsf C(F-,Y_1) \to \lim \mathsf C(F-,Y_2)$$.