\[def:subobclass\] in App. \[apx:topos\])]{}.

Given arrows $\alpha,m$ as in the diagram in Definition \[def:fpbc\], we can construct the most general pullback, called final pullback complement [@DT87; @chhk:sesqui].

\[def:fpbc\] A pair of arrows $I 
    \oset{\gamma}{\to} F 
    \oset{\beta}{\to} G$ is a *final pullback complement (FPBC)* of another pair $I 
    \oset{\alpha}{\to} L \oset{m}{\to} G$ if

-   they induce a pullback square

-   for each pullback square $G \oset{m}{\leftarrow} L \oset{\alpha'}{\leftarrow} I'
          \oset{\gamma'}{\to} F' \oset{\beta'}{\to} G$ and arrow $f \colon I' \to I$ such that $\alpha \circ f = \alpha'$, there exists a unique arrow $f' \colon F' \to F$ such that $\beta \circ f' = \beta'$ and $\gamma \circ f = f' \circ \gamma'$ both hold (see the diagram to the right).

;

Final pullback complements and subobject classifiers are closely related to partial map classifiers (see[ Def. \[def:partialmap\] in App. \[apx:topos\] and]{} [@DT87 Corollary 4.6]): a category has FPBCs (over monos) and a subobject classifier if and only if it has a partial map classifier. These exist in all elementary topoi.

\[prop:fpbc-soc\]

Let $\C$ be a category with finite limits. Then the following are equivalent:

(1)

:   $\C$ has a subobject classifier $\mathtt{true} \colon \mathbf{1} \rightarrowtail \Omega$ and final pullback complements for 