$\newcommand{\C}{\mathcal{C}} \newcommand{\I}{\mathcal{I}} \newcommand{\L}{\mathcal{L}} \newcommand{\Hom}{\mathrm{Hom}_\C} \newcommand{\op}{\mathrm{op}} \newcommand{\colim}{\mathrm{colim}}$
Let $\C$ be a locally small category and $(\I,X)$ be a diagram in $\C$. That is, $\I$ is a small category and $X$ is a covariant functor. I know from textbook, that if colimit of the diagram exists, then there is a natural isomorphism
$$ \lim_{i \in \I^\op} \Hom(X_i,\cdot) \iff \Hom(\colim_{i \in \I} X_i,\cdot) $$
Which can be indurstood as representation of compound contravariant functor by the colimit existing in $\C$.
I wonder that if limit of $(\I,X)$ exists in $\C$, then this result must dualize to
$$ \colim_{i \in \I^\op} \Hom(X_i,\cdot) \iff \Hom(\lim_{i \in \I} X_i,\cdot), $$
And I decided to prove this result myself, but I have some problems. Now I think that this is not a correct dualizartion of the firs statement, but I still want prove it, as colimits still might have some representation.
Is the second statement correct? How to prove it?
My work so far:
Denote by $(L,\lambda)$ limit of $(\I,X)$ in $\C$.
Let $A$ to be any object in $\C$. Then colimit $(\L^A,\Lambda^A)$ of diagram $(\I,\Hom(X,A))$ exists in the category $\mathsf{SET}$ as $\mathsf{SET}$ is cocomplete.
In order to prove natuaral correspondence I need to construct a bijection $\alpha_A$ between $\L^A$ and $\Hom(L,A)$ and then prove that it's natural. Using pullbacks $\lambda_* : f \mapsto f \circ \lambda$ as legs of $\Hom(L,A)$ defines a cocone over $(\I,\Hom(X,A))$. And by properties of colimits there is a unique map $\alpha_A : \L^A \to \Hom(L,A) $ with $\Lambda^A \alpha_A = \lambda_*$.
Now I need to find an Inverse of $\alpha_A$ and I stacked here. And I know additionally that $\Lambda^A$ is surjective and that $$\L^A \cong \frac{\coprod_{i \in \I} \Hom(X_i,A) }{(\sim)}. $$
But this seems to produce the map of the same direction.
p.s.
I defined some mathjax commands \C,\I,\L,\Hom, \op,\colim to speed-up typesetting. I think, you can use them in answers and comments