Category with zero morphisms

Yes. The idea is the same as the proof of the uniqueness of identities in a monoid. If $f_{X, Y} : X \to Y$ is a family of zero morphisms and $g_{X, Y} : X \to Y$ is another family of zero morphisms, then

$$f_{Y, Z} \circ g_{X, Y} = g_{X, Z} = f_{X, Z}$$

for every triple of objects $X, Y, Z$.