Let $I$ be a small category and $C$ a category. Suppose that $$ Hom_C ( A, lim_{ \leftarrow I} D_i) \cong lim_{\leftarrow I} Hom(A,D_i) $$ for $A \in C$ and $D: I \rightarrow C$ when the expression above makes sense. Is it true that for the same $A$ and $D$ we have $$ Hom ( lim_{ \rightarrow I} D_i, A) \cong lim_{\leftarrow I} Hom(D_i,A)?$$
I was wondering if this is true, but I couldn't figured out a solution. Could someone give me a hint or an answer?
First off: both your statements, given the existence of the (co)limits in question, are true and easy to find in standard literature. Furthermore, assuming that we only know the first statement, we can conclude the second by applying the first to the opposite functor $D^\text{op}: I^\text{op} \to D^\text{op}$:
Taking the opposite category converts limits into colimits and vice versa. I.e., denoting the same objects in the opposite category with $D_i^\text{op}$, we have $\lim_{\rightarrow I^\text{op}}(D_i^\text{op}) = (\lim_{\leftarrow I} D_i)^\text{op}$, or in more modern notation, $\lim D^\text{op} = (\text{colim} \ D)^\text{op}$, where by $(\lim D)^\text{op}$ I mean the corresponding cocone in the opposite category. Thus the second isomorphism is derived as follows:
$$ Hom_C ( lim_{ \rightarrow I} D_i, A) = Hom_{C^\text{op}}(A, lim_{ \leftarrow I^\text{op}} D_i^\text{op}) \cong lim_{ \leftarrow I^\text{op}} Hom_{C^\text{op}}(A, D_i^\text{op}) = lim_{\leftarrow I} Hom_C(D_i,A) $$
So no, existence of $\lim D$ does not imply existence of $\text{colim} \ D$, but only of $\text{colim} \ D^\text{op}$. But if it exists, this shows that your second isomorphism directly follows from the first.