The following question is taken from Arrows, Structures and Functors: The Categorical Imperative by Arbib and Manes.
Definition: Let $\textbf{K}$ be a category with zero object $0$. Then the kernel $u: K \to A$ of $f: A \to B$ is given by the pullback diagram $$\require{AMScd} \begin{CD} K @>>> 0 \\ @V{\large u\,}VV @VVV \\ A @>>{\large f}> B \end{CD}$$
What I would like to know is what does the map $0 \to B$ in the pullback diagram above mean? Is it like the zero map $f: X \to Y$ is defined as $f(x) = 0$ for all $x \in X$ or does it mean that it is a zero morphism (not sure if there is such a term).
I have seen it also defined this way in the context of exact sequence. Also, I often just see it as $0 \to B$ without any letter assign to it denoting the name of the map.
Thank you in advance.