A problen in category theory:
Let $I$ be a small category and $\mathcal C$ be a comolete category, and consider the functor $\lim : Func(I, \mathcal C) \to \mathcal C$. Show that $\lim$ is right adjoint to the functor $G: \mathcal C \to Func(I, \mathcal C)$ that takes an object to the constant functor and a morphism to the corresponding natural transformation between constant functors.
While this looked like a simple exercise, I am stuck on it.
Let $X, Y \in \mathcal C$, and $F$ a functor from $I$ to $\mathcal C$.
We have to show that there is some isomorphism $\phi : Hom(F_X, F) \cong Hom(X, \lim F)$ where $F$ is a functor from $I$ to $\mathcal C$ and the image of $F_X$ consists of only the element $X$ and the identity morphism. Of course, by definition of $\lim F$ there is one and only one morphism from $X$ to $\lim F$, so $\phi$ is directly determined. However, I am confused as to why $\phi$ has to be an isomorphism and natural in both components. Indeed, there can be a lot of natural transformation from $F_X$ to $F$... am I missing something? I showed that such a $\phi$ must saitsfy $\phi(\mathcal E \circ \eta^f) = \phi(\mathcal E) \circ f$ where $\mathcal E$ is a natural transformation from $F_Y $ to $F$.
It seems like I have to somehow show that there is only one possible natural transformation from $F_X$ to $F$...