\begin{definition}[Definition:Limit of Sets (Category Theory)]
Let $\II$ be a small category.
Let $D : \II \to \mathbf {Set}$ be a diagram in the category of sets $\mathbf {Set}$.
The limit of sets of $D$ is defined as:
:$\lim D := \set {\family {a_i}_{i \mathop \in \II}: \paren {\forall i \mathop \in \II : a_i \in \map D i} \wedge \paren {\forall i, j \mathop \in \II : \forall f \in \map {\mathrm {Hom}_{\II} } {i, j} : \map {\map D f} {a_i} = a_j} }$
The corresponding projections $\pi_j : \lim D \to \map D j$ are defined as:
:$\forall j \in \II: \map {\pi_j} {\family {a_i}_{i \mathop \in \II} } := a_j$
\end{definition}