The fact that deduction theory can be transformed (cf. §4) in the theory of construction of exact squares means that (cf. §2) exact squares are the building blocks to describe relational composition, that deduction is the study of a category of relations and, in particular, the determination of triples of relations $R : X \pto Y$, $S: Y \pto Z$, $T=X\pto Z$ such that $S\circ R=T$.

Since in the case of $\Set$, the relational composition of $R,S$ is obtained as a union of subobjects of $Z$, deduction theory can still be defined as the description of the structure of “objects of subobjects” $PZ$. In general we argue that since $PZ$ is the free complete lattice over $Z$ (an observation true in $\Set$, which remains valid in every topos taken as an ambient category). I have observed that the structure of $PZ$ can also be analyzed as that of an extensor (see (Guitart, 1980) an §7). Either way, the structure of $PZ$ is that of a free $P$-algebra, and it can thus be described once we know the structure of $P$. I have suggested in (Guitart, 1982) that we can elucidate such structure as that of an algebraic universe strong enough to perform all construction of first order mathematics, and pliable enough to be valid when the ambient category is a general topos, or some category of fuzzy sets.