left exact functor
Functors come in degrees of friendliness toward exact sequences. A left exact functor is one that respects the left-hand part of a short exact sequence — it preserves the existence of kernels and the injectivity at the start — but may fail at the right-hand end, dropping surjectivity. Picture a translation that faithfully renders the beginning and middle of a sentence but garbles the final word: the failure is concentrated at one end, and that concentration is exactly what makes it measurable.
Formally, an additive functor F (covariant) is left exact if for every short exact sequence 0 -> A -> B -> C -> 0 the sequence 0 -> F(A) -> F(B) -> F(C) is exact. Equivalently F preserves all finite limits, in particular kernels and finite products. The canonical example is Hom(M, -) for a fixed module M: applying it to a short exact sequence preserves injectivity and exactness in the middle but the induced map Hom(M, B) -> Hom(M, C) need not be surjective. A contravariant functor like Hom(-, M) is called left exact if it turns short exact sequences into 0 -> F(C) -> F(B) -> F(A).
The defect of right-exactness is exactly what right derived functors repair: F is left exact precisely so that R^0 F = F and the higher R^n F splice on to form a long exact sequence. Dually, a right exact functor (the tensor product is the model) preserves the right end and is corrected by left derived functors. A functor that is both left and right exact is exact, and its derived functors vanish in positive degrees.
Apply Hom(Z/2Z, -) to 0 -> Z -> Z -> Z/2Z -> 0 (first map times 2). We get 0 -> 0 -> 0 -> Z/2Z, since Hom(Z/2Z, Z) = 0 but Hom(Z/2Z, Z/2Z) = Z/2Z. The left part stays exact; surjectivity onto the last term fails, and the missing piece is Ext^1.
Hom is left exact; the failure of right-exactness is measured by Ext.