The Stacks project

Definition 4.17.1. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is cofinal in $\mathcal{J}$ or that $H$ is cofinal 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 $y \to H(x)$, and

2. given $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x, x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $y \to H(x)$ and $y \to H(x')$ 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 $y \to H(x_ i)$ in $\mathcal{J}$ with $y \to H(x_0)$ and $y \to H(x_{2n})$ the given morphisms such that the diagrams

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

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

In other words, fixing an object $y$ of $\mathcal{J}$ consider the set $S$ of pairs $(x, b)$ where $x$ is an object of $\mathcal{I}$ and $b : y \to H(x)$ is a morphism. Consider the equivalence relation on $S$ generated by $(x, b) \sim (x', b')$ if there exists a morphism $a : x \to x'$ with $b' = H(a) \circ b$. Then $S$ should consist of exactly one equivalence class.