lim functor

Let \(\mathcal{C}, \mathcal{J}\) be categories and let \(\mathrm{FunLim}(\mathcal{J}, \mathcal{C})\) be the full subcategory of the functor category \(\mathrm{Fun}(\mathcal{J}, \mathcal{C})\) such that for each diagram \(J \in \mathrm{Ob}(\mathrm{FunLim}(\mathcal{J}, \mathcal{C})\) the limit \(\mathrm{lim}_J\) exists.

Then the limit functor is defined as covariant functor

\begin{align*} \mathrm{lim}: \mathrm{FunLim}(\mathcal{J}, \mathcal{C}) \rightarrow& A \\ (\mathcal{F}: \mathcal{J} \rightarrow \mathcal{C}) \mapsto& \mathrm{lim}_{\mathcal{F}} \end{align*}

dual to colim functor