Let $\mathcal A$ be an additive category. Then there is a free abelian category $F(\mathcal A)$ on $\mathcal A$. I'm aware of two constructions in the literature, and I'd like to relate them.
The second construction is Adelman's. This describes $F(\mathcal A)$ as $\mathcal A^{[2]} / \sim$. So the objects are pairs $A \to B \to C$ of composable morphsims in $\mathcal A$, morphisms $(\alpha, \beta, \gamma) : (A \xrightarrow f B \xrightarrow g C) \to (A' \xrightarrow{f'} B' \xrightarrow{g'} C')$ are represented by triples of maps forming two commutative squares, and the relevant congruence is obtained when we mod out by morphisms of the form $(\delta f, f'\delta, \gamma)$ and morphisms of the form $(\alpha, \epsilon g, g' \epsilon)$ for $\delta : B \to A'$, $\epsilon : C \to B'$.
The first construction is Freyd's. This says that
$$F(\mathcal A) = Fun^\oplus_\omega(Fun^\oplus_\omega(\mathcal A, Ab),Ab).$$
Here $Fun^\oplus_\omega(\mathcal B, \mathcal C)$ is the category of finitely-presentable additive functors $\mathcal B \to \mathcal C$. Without too much fuss, one works out that $Fun^\oplus_\omega(\mathcal B, Ab) = (\mathcal B^{op})^{[1]} / \sim$, where the relevant congruence mods out by those $(\psi, \phi) : (Y' \xleftarrow{f'} X') \leftarrow (Y \xleftarrow{f} X)$ such that $\phi = f'\chi$ for some $\chi : Y' \leftarrow X$. Equivalently, one mods out by pairs of the form $(\chi f, f'\chi)$ as well as pairs of the form $(\psi, 0)$.
Thus, Freyd ends up describing $F(\mathcal A)$ as the quotient by a congruence on $(((\mathcal A^{op})^{[1]})^{op})^{[1]} = \mathcal A^{[1]\times[1]}$, the category of commutative squares in $\mathcal A$.
So that's the funny thing -- Adelman and Freyd are describing $F(\mathcal A)$ as quotients of different categories -- pairs of composable morphisms versus commutative squares. I'd like to resolve the combinatorial discrepancy.
Question: What does the canonical equivalence between Adelman's model for $F(\mathcal A)$ and Freyd's model for $F(\mathcal A)$ do?
For instance, does it lift to some kind of functor between $\mathcal A^{[2]}$ and $\mathcal A^{[1] \times [1]}$?
