functor preserving zero morphism and induced functor on chain complexes

1. Proposition

Let 20240130-functor_preserving_zero_morphism_and_induced_functor_on_chain_complexes_8bca0a22355f86c50b3e315a050fae8717462859.svg be an additive categories and be a functor preserving zero morphisms Then induces a functor on the category of chain complexes

20240130-functor_preserving_zero_morphism_and_induced_functor_on_chain_complexes_678ade9765f20abc2fa1204e73c10030a6bc6da6.svg

given by

20240130-functor_preserving_zero_morphism_and_induced_functor_on_chain_complexes_39b248dd85e3004dd722b6a033ad9f35688f1cdb.svg

and sending

20240130-functor_preserving_zero_morphism_and_induced_functor_on_chain_complexes_378ba3d938a266803e11ee95017ccc9e34bae0d3.svg

2. Proof

2.1. welldefined on objects

Note that since the zero morphism

20240130-functor_preserving_zero_morphism_and_induced_functor_on_chain_complexes_f910e241aa1194356ce058461c92e1410b550a58.svg

2.2. morphisms

Note that chain complexes are specific functors from 20240130-functor_preserving_zero_morphism_and_induced_functor_on_chain_complexes_7f618346e12b66ae50bd81bd9fa127cdee52f015.svg .

hence this follows from