Kernel and Coimage of Two Composite Morphisms in Pullback Exact Category

Let f:A→B and g:B→C be two morphisms in an Abeilian category.Then there exist:(1)Im f∩Ker g=f(Ker gf);(2)Im f+Ker g=g-1(Im gf).Uncommonly,there aren't the version of image in the pull-back exact category(C,E),but the corresponding properties are obtained.That is Ker f→Ker gf→Ker g×BCoim f and Ker g→Ker gЦDCoim f→Coim gf are E-short exact sequence,where D=Ker g×BCoim f.