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

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