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An object $X$ of a category $\mathsf{C}$ is a zero object if it is both initial and terminal, that is, if for any $Y \in \mathsf{C}$ there are unique morphisms $Y\to X$ and $X\to Y$.

We define a kernel of a morphism $f\colon X\to Y$ as an equalizer of $f$ and $g$, where $g$ is the unique morphism $X \to 0 \to Y$ for a zero object $0$.

However, what if a category has more than one zero object? I know that they are uniquely isomorphic, but how does it help? How can we speak of the essentialy uniqueness of a kernel?

It is known that two different objects $X$ and $Y$ satisfying the same universal properties are isomorphic. However, given a morphism $f\colon X\to Y$ and zero objects $0_a$ and $0_b$, being an equalizer of $f$ together with $X\to 0_a \to Y$ and being an equalizer of $f$ together with $X \to 0_b \to Y$ are two different universal properties.

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  • $\begingroup$ There's a unique isomorphism between any kernel with respect to $0_a$ and any kernel with respect to $0_b$ making all the diagrams commute. Hence, even varying the zero object, kernels are defined up to unique isomorphism. $\endgroup$
    – Christoph
    Commented Nov 22, 2018 at 11:28

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The zero morphism $X\to Y$ is already unique: There are unique arrow $0_a\to 0_b$ and $0_b\to 0_a$, inverses of each other.
Also, by uniqueness $X\to 0_b\ =\ X\to 0_a\to 0_b$ and $0_b\to Y=0_b\to 0_a\to Y$.
Putting these together, we see that the two compositions $X\to 0_a\to Y$ and $X\to 0_b\to Y$ are equal.

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