Let \(\mathcal{C}, \mathcal{D}\) be categories, \(X \in \mathrm{Ob}(\mathcal{D})\) an object and \(V: \mathcal{C} \rightarrow \mathcal{D}\) a functor
An initial morphism from an \(X\) to \(V\) is an object \(P \in \mathcal{C}\) and morphism \(\phi: X \rightarrow V(P)\) such that:
- for an object \(Y \in \mathcal{C}\) and a morphism \(\kappa: X \rightarrow V(Y)\)
- there exists a unique morphism \(i: P \rightarrow Y\)
- such that: \(\kappa = V(i) \circ \phi\)