I encountered a problem when I was reading Chapter Two of the book Categories and sheaves.
In this book, for $\alpha:I\rightarrow\mathcal{C}$, the colimit $\lim\limits_\rightarrow\alpha$ is defined as a functor $ X\mapsto \lim\limits_\leftarrow Hom_\mathcal{C}(\alpha,X)$. Let $k$ be the Yoneda functor $X\mapsto Hom_\mathcal{C}(X,-)$.
Then by Yoneda's lemma, on the one hand, we have $$ \begin{align} (\lim\limits_\rightarrow\alpha)(X)&=\lim\limits_\leftarrow Hom_\mathcal{C}(\alpha,X)=\lim\limits_\leftarrow Hom_\mathcal{C^*}(k(\alpha),k(X)) \\ &=Hom_\mathcal{C^*}(\lim\limits_\rightarrow k(\alpha),k(X))=(\lim\limits_\rightarrow k(\alpha))(X)\\ &=\lim\limits_\rightarrow [k(\alpha)(X)] \end{align} $$
since colimits exist in $\mathcal{C^*}:=Fun(\mathcal{C}\rightarrow Set)$. The last equality is by Proposition 2.1.6.
One the other hand, however, $$ \begin{align} (\lim\limits_\rightarrow\alpha)(X)&=\lim\limits_\leftarrow Hom_\mathcal{C}(\alpha,X)=\lim\limits_\leftarrow [k(\alpha)(X)] \end{align} $$ by the definition of $k$. Could anyone tell me where is the mistake? Thanks for any help in advance.
Your mistake is that they define $\mathcal C^\ast$ as the opposite category of what you have. Only then $k$ becomes covariant and only then the equality $\hom(\alpha, X)=\hom(k\alpha, k(X))$ holds. But then the colimit in the end is actually a limit in the category of sets, and both calculations will agree on the outcome.