At around minute 55 in this lecture by Andre Joyal, he claims that the category $Fam(Set_{*})$ of families of pointed sets is equivalent to the category $[I_{\sharp},Set]$ of copresheaves on the walking-arrow-equipped with a section $I_{\sharp}$ (background is given below). In the video he constructs a functor $F:[I_{\sharp},Set]\rightarrow Fam(Set_{*})$. I can see how this is essentially surjective, but at the moment cannot see how it is fully faithful (this is probably quite trivial). Can someone either demonstrate the full and faithfulness of $F$ or construct the weak inverse of $F$?


Given a category $C$, $Fam(C)$ is the category whose objects are pairs $(I,\alpha)$, where $I$ is a discrete category and $\alpha:I\rightarrow\mathcal{C}$ is a functor. A map $(I,\alpha)\rightarrow(J,\beta)$ is specified by a pair $(f,f_{\star})$, where $f:I\rightarrow J$ is a functor and $f_{\star}:\alpha\Rightarrow \beta\circ f$ is a 2-cell. Equivalently, this is a collection of maps $f_{\star}(i):\alpha(i)\rightarrow\beta(f(i))$ for all $i\in I$.

The walking-arrow-equipped-with-a-section $I_{\sharp}$ is the category with two objects $[0], [1]$ and two non-trivial maps $s:[0]\rightarrow[1]$, $r:[1]\rightarrow[0]$ such that $r\circ s=\mathrm{id}_{[0]}$.


If $\{X_i\}_{i \in I}$ is a family of pointed sets, then there is a function of (ordinary) sets $X = \sum_{i \in I} X_i \to I$ which maps every $x \in X_i$ to $i$, and this has a ready-made section $I \to X$ which sends $i \in I$ to the basepoint of the pointed set $X_i$. This describes the object part of the functor $Fam(Set_\ast) \to [I_\sharp, Set]$ that is pseudo-inverse to the functor which assigns to a copresheaf $(r: X \to I, s: I \to X)$ the family of pointed sets $\{X_i = r^{-1}(i)\}_{i \in I}$ where the basepoint of $X_i$ is $s(i)$. I believe the equivalence is straightforward from here.

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