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Exactness at $Y$ means that the image of the incoming map $p$ equals the kernel of the outgoing map $Y\to 0$. The kernel of the zero map $Y\to 0$ is all of $Y$.

Conditions: $X \xrightarrow{i} Z \xrightarrow{p} Y \to 0$ is an exact sequence.; Exactness is defined as $\operatorname{im}(\text{incoming}) = \ker(\text{outgoing})$ at each internal object.

Exactness at $Y$ means that the image of the incoming map $p$ equals the kernel of the outgoing map $Y\to 0$. The kernel of the zero map $Y\to 0$ is all of $Y$.

Conditions: $X \xrightarrow{i} Z \xrightarrow{p} Y \to 0$ is an exact sequence.; Exactness is defined as $\operatorname{im}(\text{incoming}) = \ker(\text{outgoing})$ at each internal object.

In this video, a sequence of modules is exact if, at each middle object, the image of the incoming map equals the kernel of the outgoing map. For the sequence $X \xrightarrow{i} Z \xrightarrow{p} Y$, exactness at $Z$ means $\operatorname{im}(i) = \ker(p)$.

Conditions: The sequence consists of module homomorphisms.; Exactness is checked at each internal object (not the ends, unless specified by zeros).

In this video, a sequence of modules is exact if, at each middle object, the image of the incoming map equals the kernel of the outgoing map. For the sequence $X \xrightarrow{i} Z \xrightarrow{p} Y$, exactness at $Z$ means $\operatorname{im}(i) = \ker(p)$.

Conditions: The sequence consists of module homomorphisms.; Exactness is checked at each internal object (not the ends, unless specified by zeros).