I'm sorry for the very silly doubt.
Recall that if we have two subobjects $[h_1],[h_2]$ of a given object $C$, their intersection $[h]$ has the following properties:
$h=h_1 \circ t_1=h_2 \circ t_2$ for some $t_1,t_2$;
if $f: A \to C$ has the property that $f=h_1 \circ s_1=h_2 \circ s_2$, then there exists a morphism $s$ for which $f=h \circ s$.
I want to dualize 1)-2) in order two get the union of two subobjects. In such a case, I get
1d) $h \circ t'_1=h_1$ and $h \circ t'_2=h_2$ for some $t'_1,t'_2$;
2d) if $f: C \to A$ has the property that $f \circ s'_1=h_1$ and $f \circ s'_2=h_2$, then there exists a morphism $s$ for which $f \circ s=h$.
Is property $2d)$ correct?
There is alot of context missing, but still I think your 2d is not correct. It seems to me that it does not provide the right universal property for the union of subobjects. In fact, I don't understand how $f\circ s_1' = h_1$ is even supposed to work, since $f$ has codomain $A$, while $h_1$ has codomain $C$ (I read it as one of the inclusions representing the subobject $[h_1]$...).
The right way to phrase the intersection and union of subobjects is by saying that they are the meet and join in the subobject lattice. In the case of the intersection you can also say that the intersection is obtained as the pullback $$\begin{array}{ccc} A_1\cap A_2 & \overset{t_1}\rightarrow & A_1\\ {}_{t_2}\downarrow\;\;&&\;\;\;\downarrow_{h_1}\\ A_2&\underset{h_2}\rightarrow&C \end{array}$$ in your category. For the union one might be tempted to consider the pushout $$\begin{array}{ccc} A_1\cap A_2 & \overset{t_1}\rightarrow & A_1\\ {}_{t_2}\downarrow\;\;&&\downarrow\\ A_2 &\rightarrow& A_1\cup A_2 \end{array}$$ and indeed you will obtain a map $A_1\cup A_2 \rightarrow C$ from the universal property of a pushout. However it is not clear (and false for a generic category) that this morphism is monic and exhibits $A_1\cup A_2$ as a subobject of $C$. This condition is necessary and sufficient to be able to describe the union of subobjects as a pushout.
Long story short: the union of subobjects is formally dual if constructed via its universal property in the subobject lattice. It is not formally dual, when constructing it inside the ambient category.