An initial object of a category $\mathcal{C}$ is an object $I$ in $\mathcal{C}$ such that for every object $X$ in $\mathcal{C}$, there exists precisely one arrow $I → X$.
Let $\mathcal{X}$ and $\mathcal{A}$ be categories and let $U:\mathcal{X} \rightarrow \mathcal{A}$ be a functor. Let $A$ be an object of $\mathcal{A}$ and define $(A\downarrow U)$ to be the category with
- Objects: pairs $\langle X, h: A \rightarrow UX \rangle$ where $X$ is an object of $\mathcal{X}$ and $h$ is an arrow of $\mathcal{A}$
- Arrows: f: \langle X, h: A \rightarrow UX \rangle \rightarrow \langle X', h': A \rightarrow UX' \rangle given by arrows f: X \rightarrow X' of $\mathcal{X}$ such that (Uf)\circ h = h' in $\mathcal{A}$.
Question: If there is no initial object in the category $\mathcal{X}$ can there still be an initial object in the category defined above?