Limit

Remarks

Equivalently, a limit of $D$ is a terminal object of the category of cones over $D$. So a limit is unique up to a unique isomorphism of cones. When $J$ is empty, a cone is just an object, and a limit of the empty diagram is a terminal object of $\mathcal{C}$.

Example: in the category of sets, every diagram $D$ with $J$ small has a limit. It is the subset $$L = \set{(x_j)_j \in \textstyle\prod_j D(j) : D(u)(x_j) = x_k \text{ for all } u \colon j \to k}$$ of the Cartesian product of the sets $D(j)$, with $\varphi_j$ the coordinate projections. A cone $(N, \psi)$ factors through $L$ by $h(n) = (\psi_j(n))_j$, which lies in $L$ by the cone condition, and $h$ is forced by $\varphi_j \circ h = \psi_j$.

Non-example: take the diagram of two non-empty sets $X$ and $Y$ with no arrows. The cone $X \times Y \times \{0, 1\}$ with the projections to $X$ and $Y$ is not a limit. The cone $X \times Y$ factors through it in two ways, by $(x, y) \mapsto (x, y, 0)$ and $(x, y) \mapsto (x, y, 1)$, so uniqueness fails.