Projectivity for, say, modules is this: an R-module A is projective iff for any two R-modules B and C and two homomorphisms g:A\longrightarrow C and f:B\longrightarrow C where f is onto, there exists a homomorphism h:A\longrightarrow C such that f\circ h=g.

Is there a name for a property defined in the same way except that we substitute A for both B and C? What about the same change in the definition of injectivity?