Initial arrow in an arrow category Let $\mathcal{C}$ be a category. It is obvious that if $\mathcal{C}$ has an inital objet, then so does $\mathcal{C}^{\rightarrow}$ (the Arrow category of $\mathcal{C}$). I'm curious to know whether the converse is true? I.e. if $\mathcal{C}^{\rightarrow}$ has an inital object, then so does $\mathcal{C}$?

Consider the functor $\mathsf{cod}:\mathcal C^\to \to \mathcal C$ which maps the arrow $f:A\to B$ to $B$. This is left adjoint to $id_{(-)}:\mathcal C\to\mathcal C^\to$, the functor which takes an object and sends it to the identity arrow on that object. Thus $\mathsf{cod}$ preserves initial objects, so if $\mathcal C^\to$ has an initial object so does $\mathcal C$.