Projectivity can be defined by a lifting property, exactness of Hom, or splitting from a free module. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.
Definitions first
An \(R\)-module generalises both vector spaces and abelian groups. A map \(f:M\to N\) is understood through its kernel, image, and cokernel, which exact sequences place on one line.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
A small case in full
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
The reusable statement
The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.
A nearby false statement
Tensor product is right exact but not generally left exact, while \(\operatorname{Hom}_R(P,-)\) is exact precisely when \(P\) is projective.
With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.