Let $\mathcal{C}$ be a category, and let $\mathcal{J}$ be a small category (often called an **index category** or **shape**). A **diagram** in $\mathcal{C}$ of shape $\mathcal{J}$ is simply a [[Functor |functor]] $D \;:\; \mathcal{J} \longrightarrow \mathcal{C}$ - For each object $j \in \mathrm{Ob}(\mathcal{J})$, we have an object $D(j) \in \mathrm{Ob}(\mathcal{C})$. - For each morphism $\alpha: j \to j'$ in $\mathcal{J}$, we have a morphism $D(\alpha) \;:\; D(j) \longrightarrow D(j')$ in $\mathcal{C}$. A diagram is often depicted by placing the objects $D(j)$ in $\mathcal{C}$ according to the pattern given by $\mathcal{J}$, with arrows $D(\alpha)$ between them whenever $\alpha$ is a morphism in $\mathcal{J}$. Note: Diagrams are just functors. The functor doesn't have to be faithful, full, etc.