 Left lifting property against monomorphisms

A map $e:X\to Y$ has this property if every commutative square with $e$ as its left side and a monomorphism $m:A\to B$ as its right side has a diagonal filler $t:Y\to A$. The filler is unique because $m$ is monic. This definition does not itself stipulate that $e$ is an epimorphism; binary products imply that conclusion by testing against the categorical diagonal.