Learn Finnish: Aion hakea liidun. - I intend to fetch the chalk.

In predicate chaining, the infinitive (hakea) inherits its object rules from the main verb (aion). Because the main verb has a normal personal subject (the implied 'I' from the -n ending), the total object takes the standard genitive-accusative ending -n. If the main verb lacked a personal subject (for example, in the imperative Hae liitu!), the object would take the nominative form instead.

Why does aion chain with the first infinitive (hakea) instead of the third infinitive (hakemaan)?

Verbs expressing intention, will, or capability—like aikoa, haluta, and voida—act as modal verbs and always chain with the basic first infinitive. You would only use the third infinitive (the -maan / -mään form) if the main verb expressed motion (like mennä) or the beginning of an action (like ruveta).

Does hakea always take a total object when it means 'to fetch'?

Not always! It depends entirely on whether the object is a complete, countable thing or an uncountable mass. Here, you intend to fetch a single, whole piece of chalk, so it takes the total object case (liidun). If you were fetching an unspecified amount of something (like coffee), it would take the partitive case instead: Aion hakea kahvia.

I know means chalk. Just as a quick refresher, why does the change to in ?