AB5 is equivalent to exactness of filtered colimits

Assume AB5. Its defining identity [L1] is exactly the directed-subobject identity in [L5]. The forward implication of the cited equivalence therefore says that every filtered colimit functor is exact.

Conversely, assume every filtered colimit functor is exact. Let $I$ be a small directed poset indexing a family of monomorphisms $b_i:B_i\rightarrowtail A$, and let $c:C\rightarrowtail A$ represent a fixed subobject of $A$. For each $i\in I$, form the pullback square tikzcd P_i \arrow[r] \arrow[d] & C \arrow[d, "c"] \\ B_i \arrow[r, "b_i"'] & A. By [L4], the top-left leg represents $B_i\wedge C$. These pullback squares assemble into a diagram in ${\mathcal{A}}^I$. Because the filtered colimit functor ${\mathrm{colim}}_I$ is exact, [L2] and [L3] say that it preserves finite limits, so its colimit square tikzcd P \arrow[r] \arrow[d] & C \arrow[d, "c"] \\ B \arrow[r, "b"'] & A is again a pullback.

The image of $b:B\to A$ is the join ${\bigvee }_i{B}_i$: each $b_i$ factors through $b$, so every $B_i$ lies below $\mathrm{im}(b)$, and any common upper bound of the family receives $b$ by the colimit universal property, so [L4] makes $\mathrm{im}(b)$ the least such upper bound. The same argument inside $C$ shows that the image of $P\to C$ is ${\bigvee }_i(B_i\wedge C)$.

Because the square of step 1.2 is a pullback, [L4] identifies the image of $P\to C$ with the meet of the subobject represented by $\mathrm{im}(b)$ and the fixed subobject $C$. Using step 2.1, this gives $\left({\bigvee }_i{B}_i\right)\wedge C={\bigvee }_i(B_i\wedge C),$ which is exactly AB5 by [L1].

Therefore AB5 is equivalent to exactness of filtered colimits.