Why is it crucial that $\kappa$ is a regular cardinal in the definition of $\kappa$-accessible categories?

Question

In the definition of a $\kappa$-accessible (or presentable) category, the cardinal $\kappa$ is always supposed to be regular.

What happens in the irregular case?

Answer 1

The following exercise is found in [Adámek and Rosický, Locally presentable and accessible categories]:

1.b Non-regular cardinals

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(2) Prove that if the concept of a $\lambda$-directed colimit is extended from regular cardinals to all infinite cardinals $\lambda$, then the following holds: if $\lambda_0$ is the cofinality of $\lambda$, then $\lambda$-directed colimits can always be reduced to $\lambda_0$-directed ones. In contrast, if $\lambda_0 < \lambda$ are two regular cardinals, verify that $\lambda$-directed colimits cannot be reduced to $\lambda_0$-directed ones: find a category with $\lambda_0$-directed colimits which fails to have $\lambda$-directed colimits.

(3) Suppose we delete the requirement that $\lambda$ be regular from the definition of a locally $\lambda$-presentable category. Prove that then a category is locally $\lambda$-presentable if it is locally $\lambda_0$-presentable, where $\lambda_0$ is the cofinality of $\lambda$.