Pushout of a module extension ($B_f=(A\prime \oplus B)/\{(f(a),-i(a))\}$)

Pushing an inclusion $i:A\to B$ along $f:A\to A\prime$ gives a new module extension with kernel $A\prime$ and the same quotient. This pushout realizes the covariance of the degree-one Ext functor in its second argument.