Some properties of pullback diagrams and pushout diagrams in abelian categories

The following results for pullback diagrams and pushout diagrams in abelian categories were obtained. Proposition 1. In abelian categories, consider a commutative diagram ･･･ where two squares are pullback diagrams and two morphisms A → C and A′ → C are monomorphic. Then each of the following statements is sufficient to ensure that P and P′ are isomorphic. (a). A → C and A′→ C are equivalent. (b). A′→ C represents the image of P → A → C. (c). P → A is epimorphic, and the image of P → A → C is equal to the image of P′→ A′→ C. (d). B → C is epimorphic, and the image of A → C is equal to the image of P´→ A′→ C. proposition 1*. In abehan categories, consider a commutative diagram. ･･･ where two squares are pushout diagrams and two morphisms C → A and C → A´ are epimorphic. Then each of the following statements is sufficient to ensure that P and P′ are isomorphic. (a*). C → A and C → A′ are equivalent. (b*). C → A´ represents the coimage of C → A → P. (c*). A → P is monomorphic, and the coimage of C → A → P is equal to the coimage of C → A′→ P′. (d*). C → B is monomorphic, and the coimage of C → A → P is equal to the coimage of C → A′→ P′