If $E$ has pullbacks then $\text{SplitEpi}(E)$ has pullbacks

Question

The book From groups to Categorical Algebra has as exercise to prove that if $E$ has pullbacks then the category of split epimorphisms of $E$ has pullbacks. This category consists of split epimorphisms along with a fixed section as objects, and the morphisms are commutative squares such that the diagrams restricted to only having the split epimorphisms and only having the sections commute. By drawing a messy diagram and trying to "chase" equal morphisms, i believe i was able to prove this. It was already a lot of work to do this on paper so for obvious reasons i won't try doing it in LaTex (if requested i can post picture). Is there an easier way to do this?

Answer 1

One way is to realize the category of of split epimorphisms in $E$ as a category of functors, and then prove the more general fact that (co)limits in categories of functors valued in a category may be computed "pointwise" from (co)limits in the category. This is the same amount of effort as the direct proof, but with a more general result, and thus more efficient.