Initial arrow in an arrow category

Question

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 $\mathcal{C}$?

Answer 1

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 $\mathcal C$.

Answer 2

If $\mathcal{C}^{\to}$ has an initial object, then so does $\mathcal{C}$. As a hint, notice that if $f : A \to B$ is any morphism in $\mathcal{C}$, then there is a morphism $\mathrm{id}_A \to f$ in $\mathcal{C}^{\to}$. Hover over the boxes below for (much) bigger hints.

It follows that, if $\mathcal{C}^{\to}$ has an initial object, then it has an initial object of the form $\mathrm{id}_I : I \to I$.

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Finally, note that given any object $A$ of $\mathcal{C}$, morphisms $I \to A$ in $\mathcal{C}$ correspond with morphisms $\mathrm{id}_I \to \mathrm{id}_A$ in $\mathcal{C}^{\to}$.