functor preserving zero morphism and preserving zero object
1. Proposition
Let 
be pointed categories and 
be a functor
TFAE:
preserves zero morphism- preserves the zero object
2. Proof
2.1. 1) 
2)
Note that 
is the zero morphism 
and is thus preserved.
Therefore 
is the zero morphism and furthermore, factors throgh the zero object
here 
is split epi and a monomorphism, thus an isomorphism (cf. monic split epi as isomorphism).
2.2. 2) 1)
Let 
be the zero morphism, then by assumption, there exists a factorization
Applying and 
results in
hence 
factorizes through 
, thus
