Definition 4.17.3. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is initial in $\mathcal{J}$ or that $H$ is initial if

1. for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exist an $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $H(x) \to y$,

2. for any $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x , x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $H(x) \to y$, $H(x') \to y$ in $\mathcal{J}$ there exist a sequence of morphisms

$x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x'$

in $\mathcal{I}$ and morphisms $H(x_ i) \to y$ in $\mathcal{J}$ such that the diagrams

$\xymatrix{ H(x_{2k}) \ar[rd] & H(x_{2k + 1}) \ar[l] \ar[r] \ar[d] & H(x_{2k + 2}) \ar[ld] \\ & y }$

commute for $k = 0, \ldots , n - 1$.

There are also:

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