U dativu pišem baki, a u akuzativu vidim baku svake nedjelje.

Because Croatian nouns change form depending on their grammatical role in the sentence.

The base form is baka = grandmother / grandma.

• baki is the dative form: to grandma
• baku is the accusative form: grandma as a direct object

So:

• pišem baki = I write to grandma
• vidim baku = I see grandma

English mostly uses word order and prepositions for this, while Croatian often shows it through case endings.

Because the person receiving the writing is treated as an indirect object.

With pisati = to write, Croatian commonly uses:

• the thing being written in the accusative
• the person receiving it in the dative

For example:

• Pišem pismo baki. = I am writing a letter to grandma.
• Pišem baki. = I am writing to grandma.

So the dative answers to whom? → baki.

Because vidjeti / vidim is a transitive verb, and its direct object normally goes in the accusative.

Here, grandma is the person being seen, so she is the direct object:

• Vidim baku. = I see grandma.

This is one of the most basic uses of the accusative in Croatian.

Good question: in u dativu and u akuzativu, the words dativu and akuzativu are actually in the locative, not in the dative or accusative.

Why? Because u here means in, and with the meaning in, it takes the locative:

• u dativu = in the dative
• u akuzativu = in the accusative

So the sentence is partly a grammar/example sentence. It is talking about the cases.

The dictionary form is baka.

Here is the singular declension:

• Nominative: baka
• Genitive: bake
• Dative: baki
• Accusative: baku
• Vocative: bako
• Locative: baki
• Instrumental: bakom

So baki and baku are just different case forms of the same noun.

A often means something like and, but, or whereas, depending on context.

In this sentence it introduces a comparison/contrast between two grammatical situations:

• u dativu pišem baki
• a u akuzativu vidim baku

So here a is best understood as and/whereas. It is not a strong contradiction, just a contrast between two examples.

Svake nedjelje is a time expression meaning every Sunday, and it uses the genitive singular.

• svake = feminine genitive singular of svaki = every
• nedjelje = genitive singular of nedjelja = Sunday

This genitive pattern is common in recurring-time expressions.

So:

• vidim baku svake nedjelje = I see grandma every Sunday

You may also hear svaku nedjelju, which is also used for every Sunday, but svake nedjelje is very natural and common.

Here it means Sunday.

In Croatian:

• nedjelja = Sunday
• tjedan = week

So svake nedjelje means every Sunday, not every week.

It is understandable Croatian, but it clearly sounds like a teaching/example sentence because of u dativu and u akuzativu.

A normal speaker would usually not say those parts unless talking about grammar. In everyday speech, you would more naturally say just:

• Pišem baki, a vidim baku svake nedjelje.

The full sentence is useful because it explicitly shows the difference between the dative and accusative forms.

Yes. Croatian can omit the written thing if it is obvious or unimportant.

So:

• Pišem baki. = I’m writing to grandma.

If you want to be more explicit, you can add the object:

• Pišem baki pismo. = I’m writing grandma a letter.
• Pišem baki poruku. = I’m writing grandma a message.

So the shorter version is completely fine.