cokernel of a zero morphism

1. Proposition

Let 20240205-cokernel_of_a_zero_object_760824120891fc757bc0445145b027d09b237baa.svg be a pointed category and the zero morphism for an object . Then the cokernel is given by the identity

Suppose there exists an object 20240205-cokernel_of_a_zero_object_ccda4f3d79b188d47d3d14d7b71cfcde42b0c6c3.svg and a morphism , then we get

since 20240205-cokernel_of_a_zero_object_5461866a7289ba4c56a10a82a5622bfbdcb4ae22.svg is the unique morphism making

commute