## weaker projectivity/injectivity

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?