Definition 4.1 (Diagram). Let $J$ be a category (almost always small, and often finite). By a diagram of shape $J$ in a category $C$, we mean a functor $D:J\to C$. The objects $D(j)$, $j\in \mathrm{ob}J$ are called vertices of $D$, and morphisms $D(\alpha )$, $\alpha \in \mathrm{mor}J$ are called edges of $D$.