Induced morphisms into pullback The induced morphism by the universal property of pullback,  when is it an epimophism ( I'm looking at it in a regular category, when it's induced from a coproduct )
 A: I'm not sure this is quite the answer you are looking for, but in the case of the category of Sets, one can say:
Notation: Consider the case of sets $X$ with maps to some fixed set $S$ that you want to take the fiber product over, and let $p_X$ denote the map $X\to S$. Then given a test object $T$ with $f_A: T\to A$ and $f_B: T\to B$ having $p_Af_A = p_Bf_B$, there is an induced map $T \to A\times_S B$. 

(i) you can check surjectivity of $T \to A\times_SB$ on the fibers over $S$, reducing to the case when $S$ is a point. The fiber product over a point is just the usual product.

In other words, $T \to A\times_SB$ is surjective if and only if for every $s \in S$, the map $$p_T^{-1}(s) \to p_A^{-1}(s)\times p_B^{-1}(s)$$ is surjective.

(ii) when $S$ is single a point, surjectivity of $T\to A\times B$ happens when $A\times B \subseteq T$ (but not naturally).

More precisely, there needs to be an injection $A\times B \to T$ commuting over $A$ and $B$. This is just because a map of sets $T \to A\times B$ is surjective if and only if there is a (not natural) choice of section $A\times B \to T$.
Remarks:
There won't be any nice criterion for this in terms of surjectivity of $f_A, f_B$ - consider the case $A = B$ and $f_A = f_B$, where the map above on fibers is the "diagonal" map, which is never surjective unless $A$ is a point, or empty. Rather, the fiber product is a surjection when the maps $f_A, f_B$ are "enough like an orthogonal pair of projections" (at the fiber level).
If your category is not Sets, but rather some sort of sets with structure which is (equivalent to) a subcategory of Sets, then the above will still apply as long as epimorphisms $=$ surjections (although the section $A\times B \to T$ will probably fail to be a morphism in your category). This discussion will not apply in a category like Rings where epimorphisms $\not=$ surjections.
