At the best of my possibilities, my original question should have sound like this. Assume we have a category $C$ in which objects are sets (so I may speak about surjectivity and injectivity of morphisms in the elementary sense) and with small limits and assume we have a pair of morphisms $f,g:a\to b$ in $C$ admitting a common section $s:b\to a$. Assume also that they admit a coequalizer $p:b \to c$ and consider the kernel pair $b\times_cb$ of $p$. Then it is clear that there exists a unique morphism $f\times_cg:a\to b\times_cb$, the comparison morphism, given by the universal property of the kernel pair.
Question: Is $f\times_cg$ surjective?
I know that without the hypothesis on the common section, I could not expect this to be true. In Set, for example, the kernel pair of the coequalizer should be the smallest equivalence relation $R$ on $b$ such that $(f(\cdot),g(\cdot))\in R$. But what about in this more particular case?
No, there is no reason for this. In Set, a special case of your question is whether every reflexive relation is an equivalence relation!