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Why any composite with a zero arrow must itself be a zero arrow? I interpret this as

$a \rightarrow z \rightarrow b \rightarrow c = a \rightarrow z \rightarrow c$

(the zero arrow in the composite is $a \rightarrow z \rightarrow b$)

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I assume $z$ is a zero object. That is $z$ is initial and terminal. Being initial, for all object $c$ there is a unique arrow $z \to c$, so for any arrow $b \to c$, $$ (z \to b \to c) = (z \to c) .$$ Being terminal, for all object $a$ there is a unique arrow $a \to z$. Composing (on the left) by this arrow for some $a$, you get $$ (a \to z \to b \to c) = (a \to z \to c).$$

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